According to Moviemistakes.com eagle-eyed Harry Potter fans have spotted 15 errors in Harry Potter and the Prisoner of Azkaban with most of the errors in the movie related to the continuity of the film. The film has only been in release for one week, so except additional fans to try & spot more errors.
According to CBBC Newsround:
One post mentions a scene in Lupin's office. The Marauder's Map is placed on a table, but then it disappears!
Another fan says that when Hermione punches Malfoy, she spins round and hits him all in one motion.
Some of the errors listed on Moviemistakes.com:
When Harry is attacking Malfoy, Crabbe and Goyle near the Shrieking Shack while under the invisibility cloak, Harry moves very rapidly, as he attacks two people at the same time. They couldn't possibly have been hit by Harry as he was swinging one of them around by the scarf. Also he is far away from this action and no contact is seen to have happened.
When Harry and Hermione are watching the punch scene again from a distance, you can tell that it's Hermione's body double because her hair is longer and straighter.
Moviemistakes.com has listed Harry Potter and the Philosopher's Stone as having 117 errors & Harry Potter and the Chamber of Secrets as having 136.