(at around 45 mins) When Harry asks Professor Lupin why he didn't allow him to have a chance with the Boggart, Lupin says that he was afraid that it would turn into Voldemort in front of everyone. But the Boggart has already turned into a Dementor before Lupin jumped in front of Harry to redirect the creature at him.
(at around 25 mins) In his opening lines as Headmaster, Albus Dumbledore referred to "...Professor Grubbly-Plank..." and "...his remaining limbs..."; however, in the books (and in Harry Potter and the Order of the Phoenix), the character is revealed to be *female* (Professor Wilhelmina Grubbly-Plank, as played by actress Apple Brook).
(at around 1h 2 mins) When Harry is under the invisibility cloak and throwing snowballs at Malfoy and his cronies, there is a point at which he pulls down the pants of Crabbe and kicks him. In the next shot, Crabbe who was previously on the ground is now standing up, perfectly fine, while Goyle is on the ground in the position of the first. In addition, when the view changes to a long shot, you can see one person being spun around by their scarf and another getting his hat pulled over his eyes, which would be impossible for one person to do on their own.
(at around 35 mins) When Harry is bowing to the hippogriff, his tie is done up wrong - the thin part is much longer than the front part. But when he is riding the hippogriff his tie is done up properly.
(at around 1h 25 mins) In the hut, when the rock comes flying in the window, it breaks the urn into three large pieces. When we see it broken again later (at around 1h 50 mins), it breaks into several small pieces.
(at around 1h 30 mins) When Harry uses Expelliarmus on Snape, Snape goes flying into a poster bed which crashes down around him. In a later shot, the bed is again intact, with Snape lying on it, unconscious.
(at around 58 mins) When Harry and Professor Lupin are walking through the forest discussing the Dementors, they step up to a large rock and Hedwig is sitting there waiting. All of them are clearly in the shade of the trees when they arrive at that spot but when the shot changes to a view from behind them, they are all standing in bright sunlight.
(at around 1h 40 mins) The first time Hermione howls at Lupin the werewolf to save Harry, Lupin gives a gentle howl back while running toward her. However, the second time she howls to save Harry, when they are back in time (at around 1h 55 mins), Lupin only growls while running toward her.
(at around 1h 23 mins) When Hermione punches Malfoy in the face, you see her point her wand to his neck with her right hand. When she turns around you can still see the wand in her right hand. When she actually punches him, she uses her right hand, but there is no wand in it.
(at around 43 mins) When the DADA class lines up to each have a go at the Ridikkulus Charm, Seamus is clearly behind Ron in the queue, followed by Harry. Once Ron ends his turn, Parvati is then next (not Seamus), followed by Harry.
(at around 1h 30 mins) In the Shreiking Shack, Lupin, wand in hand, stands over Sirius, who is on the floor. There is a double door in the background, and the door on the left goes from fully open to fully closed several times. Although the shack's walls are gently swaying, their movements wouldn't cause the door to open and close to that degree.
(at around 1h 15 mins) When Harry is interrogated by Professor Snape about the Marauder's Map in the darkly-lit hallway, the drawstrings on Harry's sweatshirt change position. In one shot, they fall parallel, but in the next shot, they criss-cross. Also, when Lupin is talking to Harry in his classroom minutes later, the drawstrings switch from being cris-crossed to falling parallel between shots.
(at around 34 mins) When Harry is bowing to the hippogriff and steps on a twig, Hermione jumps and holds Ron's arm. In the next shot with them in the background, you see her doing the exact same move again.
(at around 1h 45 mins) When Hermione uses the time turner with Harry, Ron is being tended to by everyone due to his injured leg. The nurse places the items Ron's foot is resting on the floor, they remain on the floor for the rest of the scene.
(at around 27 mins) When the Gryffindors first enter the Gryffindor common room after the Fat Lady has smashed her glass, her portrait swings into the common room, but in the next shot from the inside of the common room there is no sign of her portrait whatsoever.
(at around 30 mins) During the first Divination lesson, Professor Trelawney picks up Neville's cup from the table in front of Dean Thomas, comments on it, puts it back on its saucer and moves on. Immediately afterwards Neville grabs the same cup now standing directly on the tablecloth.
(at around 30 mins) When Professor Trelawney takes Harry's tea cup and looks at the tea leaves, she sets the cup down quickly in shock, onto the saucer. In the next shot, Ron is seen taking the cup which is placed on the tablecloth.
(at around 9 mins) When Stan Shunpike hands Harry his ticket on the Knight bus he gives it to him using his forefinger and middle finger in the front shot but when the shot is on his back (facing Harry) he is clearly using different fingers to hand the ticket to Harry.
(at around 1h 4 mins) When Harry and the gang are back in Hogsmeade after tormenting Malfoy, Harry is shown wearing the hood of his hoody. However, we see him a second time with the hood on and the next shot he is invisible. When he goes into the pub to listen to the conversation with Fudge, he is under the Cloak with no hood on.
(at around 16 mins) When Harry has opened the Monster Book of Monsters, Harry is laying on the bed taking off his shoe while the book is hiding under the bed. As a train passes by the building shakes and causes little piles of dirt to fall in the floor. When Harry holds his shoe over the edge of the bed and then jumps on the book, the dirt is gone.
(at around 48 mins) When Dumbledore reaches the landing where the Fat Lady had been ripped, we are put into the door's view, as he rubs the loose paper up so the screen goes black for a short time. In the next scene you can see the small camera and the camera hole on the door.
(at around 17 mins) [Widescreen edition only] In the Leaky Cauldron, when Mr. Weasley pulls Harry aside to talk to him about Sirius Black, the number 301 briefly flashes in the bottom left side of the screen below the "Have you seen this wizard?" poster.
You can sometimes see lighting devices reflected in the characters' glasses. As Harry is telling Hagrid "It'll be fine" as they are hurrying out the door, you can see lighting equipment reflected in Harry's glasses. Also, in Trelawney's first class, you can see similar lighting equipment reflected in her glasses.
(at around 12 mins) The Knight bus squeezes between two buses on Lambeth bridge while traveling North towards the Leaky Cauldron. The other two buses are heading South but one of them has Kings Cross as a destination which is North of the river and is therefore traveling in the wrong direction.
Underage wizards are not allowed to use magic away from school, which is why Harry was sent an official warning when Dobby used magic in the house in Harry Potter and the Chamber of Secrets. Yet at the start of the film, Harry is using the 'Lumos' spell to light up his wand and do his homework, something which would have seen him expelled according to his previous written warning.
(at around 1h 29 mins) While being swung around on a branch of the Whomping Willow, Hermione reaches out one hand, grabs Harry by the front of the shirt and lifts him up. That would require an impossible amount of strength especially for a 13 year old girl.
When the Fat Lady (Dawn French) abandons her portrait outside the Gryffindor Common Room, in fear of Sirius Black (Gary Oldman), she's found in an "African plains" portrait of giraffes and hippopotamuses. Hippos spend virtually their entire lives in/near the water.
When the class is on the staircase, immediately before the Fat Lady's disappearance from her picture is discovered, Devon Murray (Seamus Finnegan) can be seen looking directly at the camera for several seconds.
(at around 44 mins) After the class has finished dealing with the boggart, Harry's scar appears to be on the wrong side. We are seeing his reflection in the mirror on the door of the wardrobe that holds the boggart.
The Marauder's Map clearly reads "Mooney" rather than the correct spelling which is, as J.K. Rowling spells it "Moony." This was done deliberately as an in joke, as the film's visual effects supervisor is named Karl Mooney.
Professor Lupin says that he was watching the Marauder's Map and that the map showed Peter Pettigrew on the grounds. Had he seen this, he would have also spotted two 'Harry Potter' and two 'Hermione Granger' symbols on the grounds on the map as there was one of each from the present and one of each from the future. This however goes unmentioned. There is however a possibility, that the map shows only persons existing in its time line. Eventually, Professor Lupin might have been aware of possibilities of time travel, and/or did not need to make any mention on the fact of double existence of Harry and Hermione, being focused on Pettigrew.
(at around 21 mins) When the Dementor attacks Harry on the Hogwarts Express, everything freezes in ice. The bottle on the table does not expand like freezing liquid would. (Only water expands when it is frozen, if the liquid is alcohol of some sort, it will shrink when frozen if it is strong enough. Or maybe there is a spell on it, there is a number of possible explanations for this.)
(at around 27 mins) In the first scene with the Fat Lady, the background of the painting appears to be inside a structure, with a pillar to the left of the picture. When the picture is next seen after having been slashed (at around 48 mins), the background has completely changed, with a structure in the far distance. Hogwarts portrait backgrounds are usually static, but this change could be a result of this trauma.
(at around 1h 2 mins) While Harry (hiding under the Invisibility Cloak) is attacking Malfoy and company in Hogsmeade, he leaves no footprints behind in the snow. However, in all other Hogsmeade scenes, Harry leaves footprints in the snow.
(at around 1h 28 mins) When the Whomping Willow hits Harry and Hermione, they land on the grass hard. However, as Hermione gets up, she pushes down on the grass they are on, and you can tell that it is separate from the rest of the grass around them. They obviously landed on a softened mat rather than the regular ground.
(at around 1h 55 mins) In the later part of the movie, when a cloud moves to reveal the full moon, Harry and Hermione's faces light up as though a light shines directly in front of them. The moon however, is to their right.
(at around 48 mins) When the Fat Lady is missing from the picture, and Dumbledore is coming to examine it, Filch is close behind him. As they approach the painting, you can see Filch's cat (Mrs. Norris) with her real green eyes, instead of the red eyes the cat is supposed to have in the movie.
(at around 27 mins) As Harry and the others run up the stairs to the Fat Lady picture for the first time, you can see the actors' focus/lighting markers on the floor. They're white crosses with gaffer tape.
(at around 12 mins) During the first shot inside the Leaky Cauldron, you see a waiter make a bottle disappear. If you look closely, you can tell that not only was the bottle tipped on its side to appear as though it had disappeared, but it's still visible under the cloth used to wipe down the table.
(at around 1h 3 mins) When Harry attacks Malfoy and his cronies from underneath his Invisibility Cloak, you can see a snowflake in Harry's hair as soon as he takes off the cloak. This would not be possible as Harry has had the cloak on the entire time.
(at around 32 mins) When Neville tries to open his Monster Book of Monsters and gets attacked by it, you can see that a large amount of the paper supposedly shooting out of the book is coming from nowhere near the book.
(at around 1h 40 mins) When Sirius is lying beside the lake (with Harry beside him) after fighting with werewolf Lupin, his eyes pop open and he gasps in fear as the dementors start circling. However, when the shot changes to a from-above view, his eyes are closed.
(at around 1h 13 mins) When Harry, Ron, and Hermione are talking to Hagrid about Buckbeak's execution we see Harry get up and walk away from the rock he is sitting on. But in the next shot, Harry is sitting back down on the rock.
(at around 1h 15 mins) During the night, when Harry goes looking for the Peter Pettigrew he saw on the Marauder's Map, when Snape catches Harry, Harry puts the map in his front pocket, the shape of the map being in his pocket changes from shot to shot.
(at around 1h 30 mins) When Harry attacks Snape, the Expelliarmus spell seems to act like a Stupefy spell, established as distinct earlier in the movie. The difference between these two spells is a heavily-debated matter. It can be found in the book that both Ron and Hermione were supposed to have attacked Snape with Expelliarmus as well, which caused him to be blown backwards and knocked out. As neither Ron nor Hermione actually did attack him in the movie, this is an incorrect chain of events.
(at around 2h 5 mins) In the time-turner sequence, Hermione notes that once they go back, the time is 7:30. She gave the time-turner three turns, making their return time 10:30, if one turn is equal to one hour. When Hermione and Harry return, the clock strikes twelve times, with the last two being faint. There's nothing to say that the Hogwarts clock uses the standard chimes, so this can be interpreted as an alternate way of rendering 10:30. As such the first, louder chimes represent the hour (10) while the fainter chimes represent the quarters past the hour (2 quarters past = half past).
When the Time Turner sends Harry and Hermione back to the past, some of the re-visited events happen differently - people's positions are changed, objects are in different places, and precise movements happen in a different order. This could represent different perspectives, or a side effect of time travel.
The length of the chain used for Hermione's Time Turner varies throughout the film - sometimes it's quite long and other times it's barely longer than a normal necklace. She could have easily made adjustments to it as it suited her.
(at around 1h 6 mins) After Harry comes out of the pub when he finds out that Sirius is his god-father, just before Hermione takes the cloak off, you can see some snow on the green sheet used to make the cloak.