Awesome, Dear Boy: David Thewlis took the role of Lupin after Ian Hart assured him it was "the best part in the book" and because he was excited to make a children's film. He reportedly enjoyed filming very much.
B-Team Sequel: Chris Columbus stepped down as director due to family commitments. He stayed on as producer and then vacated the series.
Creator-Preferred Adaptation: J. K. Rowling said this was her favourite of the films, claiming she got goosebumps watching some parts of it. She also mourned the fact that she didn't think of the shrunken head herself while writing the book.
Deleted Role: Sir Cadogan, played by Paul Whitehouse, was cut almost completely out of the film, although you can still see him jumping through several paintings behind Ron, Harry and Hermione during the Fat Lady's disappearance.
Money, Dear Boy: Gary Oldman took the role of Sirius because he had not done a film in over a year and needed the work desperately. This turned out to be a positive version of the trope though as he gave a great performance, became good friends with Daniel Radcliffe while filming, and happily came back for the sequels. Combined with his role as Commissioner Gordon in The Dark Knight Trilogy, the role served as a Career Resurrection for him.
Michael Gambon replaces the late Richard Harris. Dawn French also replaces Elizabeth Spriggs as the Fat Lady.
An inversion happened with Warwick Davis. As Flitwick wasn't in the script and Cuaron wanted to use Davis anyway, he had him play the conductor for the school choir. Later films retconned this to being Flitwick anyway.
This almost happened to Dudley, as Harry Melling had lost a lot of weight in between films. They ended up just putting him in a fat suit.
Playing with Character Type: Gary Oldman is well-known for playing villains, so when he was cast as Sirius Black, a mass-murderer after Harry, no one was surprised. By the end, it's revealed that Sirius never murdered or betrayed anyone, and in later movies he becomes the closest thing Harry has to family (naturally, this came as no surprise to viewers who'd already read the books).
Romance on the Set: Sort of. At the time of filming Dawn French (the Fat Lady) was married to Lenny Henry, voice of the shrunken head on the Knight Bus.
Emma Thompson took the role of Professor Trelawney to impress her daughter.
Whether intended or not, this family friendly Harry Potter film was the very next film Cuaron directed after the sexually explicit, adults-only Y Tu Mamá También.
Troubled Production: Huge amounts of heavy rainfall nearly destroyed the sets. Helicopters had to bring bags of gravel in to prevent them from being washed away. There were numerous other weather-related difficulties from shooting on location in Scotland. Alfonso Cuaron ended up loving the resulting look of the film, however.
Alfonso Cuarón intended for the choir to show tiny people jumping on the keys. J. K. Rowling vetoed that, claiming tiny people were alien to the HP Universe. He also wanted to put a graveyard in, but she vetoed that as well - claiming it would contradict an important plot point in later books. She did however allow him to place a sundial on the grounds.
The original script called for Hermione to slap Draco as she does in the book. During a rehearsal, Emma Watson actually slapped Tom Felton with full force. She later said she had no idea why she did it, and felt horrible about it afterwards. This is why she punches him instead in the final version (as a punch is easier to fake than a slap).
Ian McKellen was asked to play Dumbledore after Richard Harris's death, but he turned it down due to the dislike Harris had always felt in real life towards McKellen (according to rumors, it all started with The Lord of the Rings, as Harris had lobbied strongly for the role of Gandalf and came to believe Ian had robbed him). Still, he later said he regretted not having played a role in the franchise.
Alfonso Cuaron wanted the Dementors to be achieved through puppetry. Once it became clear that wouldn't be feasible, he switched to CGI. However, they did use footage of puppets underwater for reference when creating the CGI.
Although Azkaban doesn't actually appear in the story, concept art of it was created - as a giant stone fortress on the edge of a large waterfall. This design was later used in Harry Potter and the Order of the Phoenix when the prison actually appears - though situated on an island in the sea, as in the book.
Lucius Malfoy would have appeared for Buckbeak's execution scene, but Jason Isaacs was unavailable.
Buckbeak can be seen pooping in the Care of Magical Creatures class. This is not the first recorded instance of a CGI animal pooing, but it is the first time it happens in a feature film.
The only Potter film to not gross over $800 million worldwide.
The only book in the series where no character (permanently) dies, and also where Voldemort does not appear in any shape or form. Unless you count Harry hearing Voldemort's laughter in a memory.