According to Harry Potter and the Order of the Phoenix, you need to know the purpose of the Room of Requirement to be able to access it. Harry and Ginny were looking for a place to hide the book, so the Room allowed them access.
(at around 1h 3 mins) During the Quidditch Match, and the celebration afterwards in the Gryffindor Common room, you can see that the students are chanting "Weasley is our king!". However, the audio track we hear is "Weasley! Weasley! Weasley!" over and over.
(at around 36 mins) During Professor Slughorn's lesson, the students are told to make a Draught of Living Death, the recipe for which "can be found on page 10". During the lesson, it is quite clear that Hermione has her book open somewhere in the middle (circa page 200).
(at around 48 mins) When Harry opens the prince's book after the scene which Ginny takes the book from him and reads aloud "This book is property of the Half-Blood Prince" the camera goes through pages of the book and there's a page in which we see "Sectumpsempra. For Enemies". The first word is "Sectumsempra" not "SectumPsempra".
(at around 1h 55 mins) After Dumbledore shows Harry Tom Riddle's diary he tells Harry the diary is a Horcrux but he should have said 'was' as the diary was destroyed in Harry Potter and the Chamber of Secrets, meaning it is no longer a Horcrux.
(at around 57 mins) During Harry's first visit to the Weasley's house in Harry Potter and the Chamber of Secrets, a close up of the clock shows an option for the family to be at the dentist. At a Slug Club meeting, Hermione has to explain to the group what a dentist does, because that occupation doesn't exist in the wizarding world.
(at around 2h 5 mins) After Harry scoops up a sip of potion from the basin, there is still a scoopful left in the basin. However, after Dumbledore drinks it, the basin appears to be empty and Harry grabs the locket from the empty basin.
(at around 1h 45 mins) When Harry and Slughorn go visit Hagrid, Hagrid sits with a wine bottle and an earthenware goblet and Slughorn has only a goblet on his side of the table. At the end of the visit, Slughorn's goblet magically appears right next to Hagrid's goblet on his side of the table.
(at around 36 mins) When Harry's potions class is brewing the Draught of Living Death, right as Seamus' potion blows up in his face, Dean Thomas' head instantly changes position, clearly as a result of a cut.
(at around 8 mins) When Dumbledore puts the Muggle house "back in order", the clock clearly shows the time as 10:35 p.m. However, the waitress Harry meets at the sandwich shop said she'd be off at "11pm" but Harry and Dumbledore see her and her companion closing up the shop before they leave the train station.
The potions book which Harry gets from Slughorn's cupboard has the words "This book is property of the Half-Blood Prince" clearly handwritten on the first page. In a later shot the handwriting changes to a more print-like font.
(at around 1h 4 mins) When Hermione runs downstairs and she's sitting with the birds flying around her we can clearly count five birds. Later when Ron comes in with Lavender and she sends the Oppugno spell to Ron, very fast we can count six birds and six birds hit the door behind Ron.
(at around 50 mins) In the Three Broomsticks pub, Slughorn spills some of his drink on the table and on Hermione. But a minute later when we see the table filmed from the side Slughorn was on, the table is dry, with no one taking out a wand to magically clean it up.
(at around 5 mins) When Harry and Dumbledore stop and stand side by side before approaching the house where Horace Slughorn is hiding, the top of Harry's head is below Dumbledore's shoulder when they show them from behind. In the next scene we see them from the front and Harry is now clearly much taller as his nose is at Dumbledore's shoulder and they do not appear to have moved.
When Harry Potter (Daniel Radcliffe spots Headmaster Albus Dumbledore (Michael Gambon) from the diner, Dumbledore is on the opposite side of the Tube (subway) tracks, and the billboard is behind the Headmaster. When Harry joins Albus, they look at the billboard that is now *across the tracks*, and Harry looks *behind* him, at the diner waitress (Elarica Johnson), instead of across the tracks.
(at around 52 mins) Harry, Ron and Hermione are walking back to Hogwarts whilst Leanne and Katie are walking in front of them, a great length between the two groups. When Leanne suddenly yells out, we see that Katie is lying on the ground, having touched the cursed necklace. A shot later we see Harry, Ron and Hermione standing almost next to Katie, whilst they should have been much further away, according to earlier shots.
In the Great Hall, when Ron (wearing his Quidditch uniform) asks Hermione about the Slug Club, Ron's right arm lies on the table in front of him in shots from behind, but in shots facing him his right arm is down beside him.
In Fred and George's joke shop Ron asks how much something is, the twins reply 5 galleons, one of the twins (closest to Ron) is leaning against the banister then in the next shot of him he is in a different position.
In the Christmas scene at the Burrow, while Harry, Lupin, Tonks, and Mr Weasley are talking about Snape: notice the trays of candies/baked goods on the table. When they are done talking and Ginny comes to sit down and give Harry a bite, a spot on the table has suddenly cleared for her to sit down her tray when Ron comes to sit in the middle of Harry and Ginny.
In Potions class, just before the students go to their cauldrons to prepare their Draught of Living Death, Hermione stands in front of Harry, but in the long shot she is instantly beside him, then she's suddenly back in front of Harry again.
In Potions class, when Slughorn removes the vial of Felix Felicis, he turns to face the class and holds the vial up for the class to see, but in the next shot facing him he holds only the very bottom point of the vial.
In the dormitory, when lovestruck Ron is going gaga over Romilda, he is wearing a brown sleeveless tee shirt and its tan trim has many holes. When Harry brings Ron to Slughorn, Ron is still wearing that tee shirt, but now some of the holes are gone and the rest of the holes have changed position.
In Potions class, when Hermione stands before the cauldron explaining what she is smelling in her closeup, we see Neville behind her with the broad space to his right (viewer's left), just as she says, "Spearmint toothpaste". In the instant cut to the wideshot, Harry is now shoulder to shoulder with Neville, too fast for him to suddenly be standing that close to Neville.
When Harry is kneeling beside Dumbledore's body, at the foot of the Astronomy Tower, Ginny joins Harry and holds him at his left side (viewer's right). In the last close-up of Professor Dumbledore's face, Harry is visible in the background with a space to his left and Ginny is now gone. That same angle was shown before Ginny joined Harry.
When Hagrid and Slughorn are drinking after Aragog's funeral, Slughorn's cup is considerably larger than Hagrid's (an effort to show size difference between Slughorn and Hagrid), and note when Slughorn holds his cup how large it is compared to his hand. But after Hagrid passes out Slughorn's cup is now the same size as Hagrid's cup - Slughorn's hand is resting on the table right beside the cup, which is now significantly smaller compared to Slughorn's hand.
When Harry and Dumbledore return from the Horcrux cave, Dumbledore sits down with his arm around Harry's neck. When Dumbledore asks Harry to go fetch Snape, his arm suddenly disappears, and there was no time for him to have moved it.
When Harry, Ron and Hermione are at the Three Broomsticks talking with Slughorn, Slughorn spills his butterbeer on Hermione and the table. Still in the conversation, the table is shown, but it's dry. Then a little later, there is foam on the table but only near Harry's glass. The rest of the table is still dry.
In the dormitory, when Harry walks in to see lovestruck Ron at the window, Romilda's card and empty chocolate wrappers are on Harry's bed. When Harry places the Marauders map and Potions book on the dresser, the card and wrappers have changed position.
When Slughorn and Harry are talking at The Three Broomsticks, Slughorn's grip on the silver mug repeatedly changes. His fingers are either at the top of the mug or at its center, depending on the camera angle.
In the living room at the Weasleys' house, while Harry, Arthur, Nymphadora and Lupin are talking, in the shot facing Lupin, Harry's shoelace is untied. After a brief cut to a shot facing Nymphadora and Arthur (where we can see Harry's hands), it cuts back to the original angle and now the shoelace is tied. (This has nothing to do with Ginny tying Harry's shoelace much later on.)
In the Great Hall before the Quidditch match, in the shots facing Hermione she is sitting opposite the empty place setting beside Ron, but in the shots facing Ron she is directly opposite him. Take note of the goblets, plate of eggs, etc.
When Dumbledore approaches Slughorn, who is disguised as a chair, both the sheer and opaque curtains' appearance changes back and forth - also note the holes or lack thereof, between Slughorn's close-ups and wideshots.
In the scene before the Quidditch match, while Ron, Hermione and Ron are talking and Ron believes that Harry put something in his drink - in some angles Ginny is sitting directly next to Harry with their arms only an inch or two apart, in other angles Ginny's arm is nowhere to be seen, when they are sitting close enough that she should be visible.
At the Burrow, when Harry is speaking with Lupin, Tonks, and Arthur, and then when Ginny joins him on the sofa, the orange blanket lying on the sofa's arm to Harry's right moves forward, back, forward, etc.
When Dumbledore and Harry first arrive in front of Slughorn's temporary residence in Budleigh, Babberton, Harry's height barely even reaches Dumbledore's shoulders (likely the actor's stand-in), but in the next shot facing them Harry's height compared to Dumbledore is back to normal, level with the Headmaster's eyes.
At the Three Broomsticks, when Slughorn and Harry speak, Slughorn holds a silver mug with a high foam head, but in the shot facing Harry, before Slughorn actually spills the drink the mug's foam head is gone (visible at the bottom of the screen), only to reappear when he spills it on Hermione ,whereupon the same previous angle is shown again without the foam.
During the first potion class scene during the movie, Hermione and Lavender are seen standing next to each other while being taught potions by Professor Slughorn. However, within the first minute or so of the lesson, Hermione and Lavender are standing far apart, then in the next shot, they are standing close together, with what looks like barely any space to move. They remain like this for the rest of the scene, but you can see the gap between the two within the first few minutes of the class.
In the Great Hall, from the time Hermione hits Ron with the book to when Harry (with bloody nose) joins them, the spoon in Hermione's bowl repeatedly faces the opposite way depending on the angle of the shot.
In the scene with Weasley's Wizarding Wheezes, as the camera pans up the front of the store, one sees the store name (in gold lettering). However, when Harry, Ron, and Hermione leave the store to follow Draco Malfoy the store name is no longer present.
When Harry finds Hermione crying at the bottom of the stairs, with the conjured canaries flying overhead, there are numerous leaves/debris on the lowest steps and near her feet, but in the shots looking down from the top of the stairs the area is clear. (This has nothing to do with the few leaves that blow about.)
After drinking the poison in Slughorn's office, Ron falls and in the overhead shots the top of his head is near the rug's edge, but in side shots his head is well below the large diamond pattern, more than a foot away from the edge.
As Harry and Luna come through Hogwart's gates, an Auror standing outside begins to walk backwards in through the gate. He makes it half-way through, but in the next shot, he is still at the end of the gate.
The very first shot of Malfoy pulling the tapestry off the Vanishing Cabinet (after tossing a green apple in his hand) is used once again later (after Harry and Luna head off to Slughorn's party), when he is pulling the tapestry off again before placing a green apple in the Vanishing Cabinet. Note the same poof of dust, angle in which tapestry falls, and Draco's body position.
When Dumbledore and Harry enter the muggle house to find Slughorn the lights are all out; the only illumination comes from their wands. They enter the hallway, then both turn towards the living room with their wands in front of them. Despite this, the wall behind them is illuminated without either of them casting a shadow. The lighting in the hallway is obviously coming from a different source, not the wands. Can't be moonlight - it is a bright white light shining directly on the wall that was not visible when Harry and Dumbledore first entered the flat.
At the Three Broomsticks, when they notice Dean and Ginny snogging, Hermione's left arm is leaning across the table in front of her, then both her arms are down at her sides in the shot facing her, but next shot, her arm is on the table once again.
When Dumbledore and Harry approach the Slughorn armchair, on the left there is a white lampshade lying on the shelf hanging over the edge. When Slughorn stands up and walks to the door to lock it, the lampshade is farther back on the shelf.
When Ginny snatches 'Advanced Potion Making' from Harry in the common room, and says "Who's the Half-Blood Prince? . That's what it says right here: 'This book is the property of the Half-Blood Prince'", she opens it at a random page in the middle of the book. When Harry sees the words 'This book is the property of the Half-Blood Prince', they are written on the first page. Also, the continuity is wrong as she is shown to switch pages between very quick shots.
When Ginny rips the Half-Blood Prince's potion book out of Harry's hand, before Harry gets up from leaning against the couch on the floor, Ginny is nowhere to be found in the chair on the very far left of the screen. As Harry walks around the couch while talking to Hermione, she is suddenly sitting in the same chair.
In the scene where Dumbledore and Harry meet Slughorn in the desolated house, some piano hammers lie on the ground. These hammers are from an upright piano but the "restored" instrument later in the scene is a grand piano.
Tied round Dumbledore's beard is a thin silver strand, with dangling silver balls, that changes position and the height of the silver balls differ in consecutive shots, depending on the camera angle throughout the film. For example, when Dumbledore and Harry first pay a visit to Slughorn.
In Potions class, when the students prepare the Draught of Living Death, as Harry tells Hermione, "Crush it, don't cut it," the way he holds/squeezes the Sopophorous Bean has instantly changed from the previous shot.
At the Burrow, when Tonks and Lupin are at the front door with Mr. and Mrs. Weasley, it cuts to an exterior shot of Harry looking out the window with his arm leaning across his knee in front of him, but in the next interior shot his arm is down at his side.
At the diner when Harry walks toward the windows, in the close-up of a window through which we see the distant platform (before Dumbledore appears), the style of the window's muntins (sash) around the glass panels is quite different in this close-up, than in the previous shots of the windows.
At the Three Broomsticks, Slughorn is holding a mug with his left hand when he speaks to Harry, but when he says, "Good to see you, Wallenby," the mug is suddenly in Slughorn's right hand, with his left arm down by his side.
In 1 or 2 scenes in the dining hall, Lavender Brown is shown sitting at a different table across from Ron Weasley. They were both in Gryffindor, so they should be at the same table. However, it is not uncommon for students to be seated at other Houses' tables. Lavender was avoiding Ron and chose not to sit with Gryffindor.
(at around 2 mins) Although the books take place in the mid-1990s, no such chronology is given in the movies, allowing the Millennium Bridge in London (built 1998-2000) to appear. On a similar note, the movies' biographical or age information about certain characters may vary from the books, but is not an internal goof.
Some have suggested Aragog is a great deal smaller in this movie than he was in Harry Potter and the Chamber of Secrets, but this only true to an extent, as spiders generally shrivel up pretty fast after dying. Add to this the fact that the persons around him this time around are quite a bit taller, plus the angles at which we saw him in his first appearance really complementing his statue while alive, and we have a very reasonable change which doesn't indicate any mistake or error by the filmmakers.
(at around 1h 55 mins) In theory, no one can Apparate into or out of Hogwarts as Dumbledore does in the last act. However, Dumbledore has secret information to override that particular magic. In fact, Harry points out this seeming problem and Dumbledore says, "Well, being me has its privileges."
(at around 8 mins) When Dumbledore first introduces Harry to Professor Slughorn, Slughorn's robe changes color more than once. However, this is due to the fluorescent lighting outside of Slughorn's home.
When Harry first has the Half Blood Prince's potions book, his signature is in the front of the book. When Ginny takes it from him, she reads his signature from the back. However, the book may have been signed at the back and the front as later in the film we see the words written in two different styles suggesting multiple instances.
Dumbledore has Riddle's diary in his possession. At the end of Harry Potter and the Chamber of Secrets, Dobby dropped the book to rescue Harry from Lucius Malfoy's wrath. The book was left on the floor, meaning that someone could return the book to the headmaster's office.
In previous films the Weasley home is surrounded by trees, while in this movie it is located in the middle of a huge empty field, so many have wanted to label this change as a goof (even if this clearly is a deliberate artistic change by the producers). However, we must bear in mind that the residents at the Burrow are a family of _wizards_! Hence, the work of removing some trees (or even transfiguring?) and boosting the growth of crop in the course of a winter shouldn't seem far-fetched at all in that context.
(at around 1h 40 mins) After Harry consumes the Felix Felicis, the vial seems to be still unused while Harry is talking to Ron and Hermione. However, when he is lowering the vial from his mouth, Harry can be seen replacing the stopper, giving the impression that the vial has not been used.
(at around 1h 45 mins) In the scene with Hagrid and Professor Slughorn, you can see them both moving their hands when talking. Slughorn's hand often disappears/seems to go right through Hagrid's arm. They were filmed separately to exaggerate Hagrid's height.
(at around 2 mins) During the destruction of the Millennium Bridge: to begin with the bridge is full of people, including a lot of school children, walking across. As the bridge begins to wobble, people start to trip up and fall down. When we see the bridge from above, as it is destroyed seconds later, there aren't any people on it.
(at around 5 mins) At the beginning when Harry and Dumbledore first disapparate, there is a full shot of them walking up to the entrance of the house. In this full shot, you can see Dumbledore's shoes sticking out of his robe as he walks, only he is actually wearing those blue bags on his shoes, the ones they wear in between takes so as not to damage the set floors.
(at around 49 mins) In the Weasley Brothers' shop scene, when they are showing Harry the Instant Darkness powder, an FX layer over the case holding the powder (meant to darken the case from a dull gray to true black) is clearly visible, extends beyond the bounds of the case and moves a bit during the shot.
(at around 36 mins) The "Draught of Living Death" page in Harry's book of "Advanced Potion Making" skips from step 8: "Chop three Valerian Roots in to small squared pieces. After cutting place it in a beaker with water. Leave it to settle for 7 minutes", to step 10: "Carefully pour the Sopophorus Beans' Juice into the cauldron making sure that ONLY the juice of 12 beans is used", without a step 9.
(at around 36 mins) The contents of Harry's book of "Advanced Potion Making" says "How to Prepare the Draught of Living Dead [sic]", whereas the actual page's heading is "Preparation of the Draught of Living Death".
(at around 55 mins) Several times throughout the movie, Harry's glasses clearly have no lens in them. They are just the frames. This is clearly evident when Harry and Ron Weasley are lying in bed and Harry is looking at the map showing the foot prints. The rest of the scenes show the glasses having lenses in them.
The goof items below may give away important plot points.
(at around 2h 10 mins) In the establishing shot right before Snape casts Avada Kedavra the tower's safety railing is clearly visible directly behind Dumbledore. But when the camera cuts to Harry's perspective of Dumbledore being hit by the spell, the railing has disappeared and Dumbledore falls right off the edge of the tower where the railing had been.
In the chase scene in the Corn field when the Death Eaters attack the Burrow, after Ginny and Harry make a stand before the Order arrive, Harry can clearly be seen pointing his wand. As the Order arrive and form the rest protective circle, he is seen pulling the wand out of his pocket.
(at around 1h 45 mins) When Hagrid is mourning over Aragog's dead body, Harry stands next to him. If you look closer, you'll notice that in faraway shots, Harry's head is clearly above Hagrid's elbow. However, in the closer shots, his head is below Hagrid's elbows.
(at around 2h 15 mins) At the end of the movie when everyone is holding their wands skyward in tribute, many have thought to see Hagrid holding someone's else's wand up. But that hand, if you look very closely, is not of Hagrid, but of a nearby student.
(at around 2h 15 mins) Near the end of the film, when Harry is lying on the ground outside Hagrid's cabin, Snape kicks Harry's wand away from his hand. When he does so, a square of neon tape is clearly seen, almost right in the middle of the shot, where Harry's wand was placed, so that actor Alan Rickman would know where to kick.