Continuity mistake: After riding down the escalator on their way to Harry's hearing, when Arthur Weasley tries to pass the turnstile, Harry's right hand reaches for the transit pass in his left jacket pocket in the side shot. Then he reaches once again in the front shot, before it cuts back to the side shot as he's still reaching for it. Also, note the man with the newspaper under his arm that passes the turnstile only to be back in line, between these shots. (00:17:10)
Continuity mistake: In scenes, but most noticeably in the court room scene, the tassle which hangs on Professor Dumbledore's hat keeps on changing position from whereabouts, and how it is hanging down. (00:22:20)
Continuity mistake: When Harry is being tested by Snape on being able to block his mind from being read, the sweat stain on his jumper (sweater) changes from larger to smaller during the scene. (01:07:00)
Revealing mistake: When the students are riding on the Thestrals to London, in one shot Luna's hair is practically still. Seeing as they're flying fast in the air, and the Thestrals' wings are flapping strongly, surely her hair would be blowing much more fiercely. (01:41:15)
Continuity mistake: After Harry enters the Gryffindor Common Room, when Seamus stands up his white shirt is untucked just a bit at his back. As he stands with his hands at his sides, in the next shot from behind Seamus, now the entire shirttail is untucked. (00:32:10)
Visible crew/equipment: After Crookshanks runs off with Fred and George's Extendable Ears the twins apparate and startle Molly, and when it cuts to Molly asking Harry if he's hungry, the actor's L-shape tape mark can be seen on the floor behind Arthur. (Only visible on fullscreen DVD). (00:15:00)
Continuity mistake: In Umbridge's office, Harry begins to write "I must not tell lies" on the paper and in the next close-up the writing on the paper differs. The paper itself moves on the pink mat, note the flowers in relation to the text. (00:37:45)
Continuity mistake: After the Weasley twins fly out of the school on their Broomsticks, when the students gather and applaud them, in the shot facing Flitwick, Nigel is standing to Harry's left, right beside him. In the shot facing Harry, Nigel is now standing a few feet away, to Harry's right. (01:33:40)
Plot hole: Harry can now see the thestral creatures because he witnessed Cedric Diggory's death in "Goblet of Fire." However, he should have been able to see the thestrals after Professor Quirrell died from Harry's touch in "Philosopher's Stone". Quirrell crumbles into dust (dying) while Harry is still conscious, watching, and fully aware of what he is seeing, thus allowing it to fully sink into his consciousness. Harry faints "after" Quirrell is dead and as Voldemort's spirit passes through his body. Nor does it only have to be a friend or loved one who dies before the thestrals can be seen - it can be anyone, including an enemy like Quirrell. There only has to be an emotional response to the death, which can be love, hate, fear, shock, etc. (In the book, Harry did not see Quirrell die, but that's irrelevant to the film).
Suggested correction: JK Rowling has said that in order to be able to see thestrals the death has to "sink in" psychologically with the person. Which was her explanation for why he couldn't see thestrals even though he witnessed his mother's death. Given Harry's age and the way Quirrell died, it is possible that Quirrell's death didn't have enough of a psychological impact on him to "sink in." After I first personally witnessed a violent death, I didn't feel the impact until a few years later. It can take time.
Continuity mistake: After being expelled, when Harry places the photograph on his desk, his wand moves closer and then farther away between shots. (00:07:15)
Continuity mistake: During Harry's first detention, before he starts writing and while Umbridge is still talking, he's sitting at the desk. He's holding the quill in his right hand, and his left hand and arm are resting on the desk. The shot switches to another angle, and his left arm is immediately off the desk. Umbridge is still talking, so there is no time lapse between the two shots. (00:37:10)
Continuity mistake: When Dolores has Sybil Trelawney out in the courtyard, Sybil's suitcase in front of her is horizontal to us, even when she bumps into it, it remains horizontal. After Professor McGonagall comes over though, the suitcase is now diagonally positioned. (00:47:35)
Continuity mistake: When Dolores Umbridge writes the "OWLs" on the board, the writing becomes smaller in later shots. (00:34:05)
Continuity mistake: In the Department of Mysteries, when Hermione says, "It's just an empty archway," the scratch on Harry's right cheek is large and bloody, but in the next shot it is significantly smaller. It continues to change in following shots. (01:48:40)
Continuity mistake: After the battle in the Hall of Prophecy, when Harry and the others land in the Death Chamber, Ginny is a good bit of distance away from Harry's feet in the side shot as they're all lying on the floor, but two shots later as they're starting to get up Ginny's head is now right beside Harry's knee. (01:52:25)
Continuity mistake: When the Order get the brooms at the start, the height they are being held at, changes during an instant cut. Harry and Kingsley's are most noticeable. (00:09:05)