When Professor Lupin is resigning due to Snape passing around his werewolf identity, Lupin gives back the Marauders map to Harry. As he points to it with his wand and says 'mischief managed' it zooms in on the map and clearly says "Mooney" rather than the correct spelling which is, as JK Rowling spells it "Moony." This was done deliberately as an in joke, as the film's visual effects supervisor is named Karl Mooney.
In the scene where 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.
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.
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.
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.
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.
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.
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.
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.
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.
When Harry, Ron and Hermione meet Malfoy's gang before reaching Hagrid's hut, Hermione is threatening Malfoy with the wand on her right hand. At the very next scene she punches him using the same hand though she holds the wand with her left one.
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,) Lupin only growls while running toward her.
During the scene 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.
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.
(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.
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.
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.
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.)
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, 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.
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.
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.
During the scene where 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.
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.
In the scene where 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.
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.
When Harry and Hermione are hiding from the werewolf, and they back away from a tree, Hermione is 2-3 steps behind Harry and closest to the werewolf. In the next shot, she is in front of Harry as he puts his arm around her to protect her.
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.
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.
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.
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.