Hermione's subplot could be interpreted as, "Doing well in school is great, but your mental health comes first. No matter how smart you are, you can still bite off more than you can chew, so if school is putting a strain on your well-being, it's ok to dial back a bit."
The incident with Lupin before the climax has the lesson of "Always remember to take your meds."
Sir Cadogan, the portrait that takes over guarding Gryffindor Tower for The Fat Lady after she's attacked by Sirius Black, and whose work style involves setting "ridiculously complicated" passwords, and changing them at least twice a day. Is he stark raving mad, or Properly Paranoid taken Up to Eleven?
Snape's behaviour in this novel, where he's at his most emotional and unhinged has drawn a lot of views. Notably fans invoke information from later series to explain why he was so personally angry at Lupin. Others however note that his taunting at Harry by using Lily's own insults about James to the latter, and then having the latter remind him, much as Lily did in The Prince's Tale that James did save his life, and repeating the same conversation, while in the finale shouting at Harry that James deserved to die because he was too arrogant to trust Sirius is more or less him vicariously trying to argue that Lily was a fool for choosing James and wondering What Does She See in Him?.
In terms of his behavior towards Black in the Shrieking Shack, did he want revenge because he believed Black was the one who sold out Lily to Voldemort? Or was he one of the Death Eaters who knew Pettigrew was the actual spy, but still blamed Black for her death because he decided not to be the Secret Keeper himself.
Did Sirius Black attempt to break into the Gryffindor dorms on Halloween night (when all the students were at the feast) because he lost track of the date? Or was he aware and chose the time knowing the place would be deserted and he could find Pettigrew unimpeded?
Ass Pull: Crookshanks being half-kneazle. Readers were already introduced to Mrs. Norris, who is a smart cat and yet isn't half-kneazle. Somewhat mollified in that it's never mentioned in the book, only brought up by Word of God much later on.
Draco in Leather Pants: Malfoy has gotten a lot of sympathy from some fans for his injury and see his parents complaints about Buckbeak being justified. This completely ignores that not only was he clearly faking the severity of his injury, but that he explicitly ignored Hagrid's instructions to the class to not insult the hippogriffs.
Ending Fatigue: Not too bad, but off to see Buckbeak get executed - diverted to shrieking shack, long Plot Dump, Werewolf ensues, time turner, going back though the evening and THEN it's over. Depending on your point of view, it either is a tense, fast-paced moment filled with story revelations or feels like too much got crammed in at the end and these should have been sprinkled throughout.
At one point in Harry Potter and the Philosopher's Stone, Scabbers bit Goyle on the knuckle (which was even referenced by Fred and George Weasley in Azkaban). Come Azkaban, and the revelation that Scabbers was actually Peter Pettigrew, who was the true person responsible for leaking Lily and James Potter's location to Lord Voldemort and thus their deaths, not to mention the revelation in Goblet of Fire that Goyle's father was most likely one of Pettigrew's comrades among the Death Eaters it becomes somewhat disturbing.
Ron carrying him around in his pocket and Scabbers even falling asleep on Harry's bed at one point in Philosopher's Stone. Counts in-universe too as when Pettigrew tries to plead with Ron for mercy, Ron (with appropriate anger and disgust) points out he let the rat sleep on his bed.
Growing the Beard: Many fans believe that this is the turning point for the series as a whole, citing the encounter at the Shrieking Shack, the darker emotions and actual feelings of loss Harry feels about his family (that had not been really explored in the previous books) and of course the Bittersweet Ending. Sirius is proven innocent to Harry and Dumbledore but remains on the run alongside Buckbeak, and Lupin has to quit, while Harry worries about Trelawney's prophecy and the consequences of sparing Wormtail potentially bringing Voldemort back. Compared to the out-and-out positive victories in the first two books, it was bleaker and prepared for the gradual darkening of the remaining books.
The part where Snape gets insulted by the Marauder's Map after Pottermore revealed that it was designed to do that to Snape and only Snape, not just anyone who tried to read it without the password.
In the novel, Hagrid expresses worry that Buckbeak might have encountered werewolf!Lupin in his escape from Hogwarts. As the movie illustrates, Hagrid need not have worried for Beaky's welfare in such a fight.
While discussing Sirius Black Hagrid says that if he had found Sirius before Pettigrew did he would have 'ripped him limb from limb'. Fudge dismisses the idea saying only a squad of top Ministry wizards would have stood a chance against him. Two books later we see a fight between Hagrid and a squad of Aurors. Hagrid wins. Easily.
Hermione's white face.note This, and a quick mention of "Hermione, very brown" at Diagon Alley when she freshly came from a vacation in the South of France, is the rare time in the entire series indicating Hermione's skin color, though several other times she is described as pale, and she is depicted as white on some front covers. Used as a justification for fans that were surprised at the casting of a black actress in Harry Potter and the Cursed Child, notably when J.K. Rowling claimed she'd never mentioned Hermione's race, even though theater casting has been color-blind like this for a while now.
"Turn to page 394."
Snape's over-the-top final rant to Fudge after Sirius escapes, "This has something to do with Potter!" is often invoked as one of his catchphrases.
Never Live It Down: Among a lot of fans, Snape never quite recovers from this book, even after re-reading the series with all his motivations revealed. He's wildly unprofessional, sadistically and openly cruel even to students like Neville to a level that would have gotten any teacher fired from a real-life school, was willing to send Sirius to the Dementor's Kiss even after he was open to turning himself in to Dumbledore, and then outed Remus Lupin as a werewolf publicly. Even the revelation from the later books about Sirius bullying Snape as a young man and Snape being loyal all along doesn't vindicate his conduct, since his incompetence, i.e. neglecting to bring or mention the Wolfsbane Potion to Remus after entering the Shack, played a major part in letting Peter escape, a detail that slipped him since he wanted to get back at Sirius and Lupin. His behaviour is so bad that the movies had to tone him down to not make him too unsympathetic.
One-Scene Wonder: Aunt Marge only appears in one chapter (which has her name on it), but still manages to be among the more despicable characters in the series, perhaps even more so than her brother Vernon.
Shipping Goggles: The series has enough of this to fill a page, but this entry probably has the best example, what has been dubbed "The Symbolic Flight." This theory states that because Harry and Hermione flew together on Buckbeak without Ron, that they are destined to be together — ignoring the fact that Ron was only not present because he broke his leg earlier, or that it's most likely there to make up for Hermione missing out on so much in Book 2, where Harry and Ron flew a car, and then had adventures while Hermione, like Ron in POA, was passed out and missing in the climax.
Accidental Innuendo: The scene from the book listed above is even more dodgy in the film, when Harry's Uncle Vernon keeps bursting into the room to find out what's going on. Vernon is played by Richard Griffiths, AKA Uncle Monty.
Author's Saving Throw: The Adaptation Expansion for the Time Turner sequence is widely agreed to be a big improvement over how it was handled in the books. A few readers missed the Foreshadowing, and thus found the twist to come really out of nowhere. The film expands on this - having Hermione suddenly turning up out of nowhere in classes at points throughout the film. Also the twist that Harry himself conjures the big Patronus is Foreshadowed by having more events influenced by the time travelers - throwing the pumpkin seeds to warn that Fudge is on the way, for example.
Base-Breaking Character: The Shrunken Head on the Knight Bus. Some find him bizarrely out of place and unnecessary, others find him a nice bit of comic relief. It's worth noting that it was approved by JK Rowling - who prevented Alfonso Cuaron from inserting a choir of tiny people playing the piano.
After everyone arrives at Hogwarts to start the new year, the Fat Lady won't let them into Gryffindor Tower because she's too busy trying to shatter a glass with her voice.
A lot of screentime is devoted to a blue bird flying through Hogwarts and twittering around; far longer than normal for a scene transition. The deleted scenes show that the sequence ran even longer - with the bird buzzing around Hagrid's head before getting shooed away.
The choir singing an arrangement of the witches' chant from Macbeth... along with croaking toads?
It should be noted that the first two here do serve some purpose in introducing the Fat Lady and the Whomping Willow tree (or just refreshing viewers' memories for those familiar with the series) so that they don't come out of nowhere when they become relevant to the plot. But the frog choir? Yeah that's just straight up Hogwarts BLAM.
Broken Base: Half the fanbase considers this one of the best (if not the best) Harry Potter film in the franchise, or at least when the Harry Potter films started getting good. The other half considers this movie to be one of the worst (if not the worst) Harry Potter film in the franchise. The film was mostly better received by critics than the first two, although it was the first HP film not to get four stars from Roger Ebert. Also, this is J.K. Rowling's favorite film. It generally comes down to either "good scripting" vs. "good cinematography" (with the film lacking in the former and bountiful in the latter), or "Darker and Edgier" vs. "whimsical and fun" (with the film relishing in the former, as opposed to the latter which had endeared audiences to the first two films).
Continuity Lock-Out: This was the point when the movies started to get hard to understand for anyone who wasn't already familiar with the respective books. Most notably, this movie never makes it clear who Padfoot, Wormtail, Moony, and Prongs are, despite later movies operating on the assumption that the viewers all know.
Critical Dissonance: The film was adored by critics when it came out, opening to the most critical success any Potter film had seen at that point. On the other side, it left the Potter fanbase fairly polarized. It was also the least successful Potter film at the box office, albeit still incredibly profitable compared to any "normal" blockbuster.
Growing the Beard: According to most professional critics, along with about half of the fans. At the very least, people agreed that the younger actors and actresses were finally fitting snugly into their roles (aside from the Narm mentioned below). What's more is that this was the first film to make deviations from the book in the name of Pragmatic Adaptation, paving the way for the future films to do so (without falling into In Name Only).
Harsher in Hindsight: Sirius's infamous line "I DID MY WAITING..TWELVE YEARS OF IT...IN AZKABAN" became this in 2018 with the release of the Allegedly Free GameHarry Potter: Hogwarts Mystery, which features an energy meter that, if empty, forces players to wait unless they buy Microtransactions.
Narm Charm: The examples in the above page, as well as "You foul, loathsome, evil little cockroach!" They're a near 50/50 mix of awesome and narm.
Never Live It Down: A lot of the fandom's complaints with Hermione stem from this film. Namely her getting Ron's line standing up to Sirius, Sirius telling her she's the smartest witch of her age instead of telling Harry he's truly his father's son, and her action scene on the Whomping Willow (despite the fact that she gets into a worse scrape than Harry in that one). If the fans ever complained about Steve Kloves turning Hermione into the Creator's Pet, expect this film to be mentioned as the root of it. This is also the movie that she starts referring to Ron as "Ronald", something she never did in the books.
At the very end of the scene where Lupin berates Harry for not turning in the Marauder's Map, there's an obvious (especially so when looking at the lighting) jump to a different take in the middle of a shot. It occurs just after Harry tells Lupin that he thinks the map might not be working since he saw Peter Pettigrew on it.
The final scene of Harry riding his new Firebolt gets a horribly smeared freeze-frame to end the film on.
Many of the film's drastic changes from the book were not welcome to some:
The excision of the explanation of the identities of Moony, Wormtail, Padfoot, and Prongs, which confused many moviegoers who hadn't read the books. Especially since it would have been relatively simple to slip into Lupin's final scene.
Ron losing his Moment of Awesome of standing up for Harry in front of a supposedly-murderous Sirius, with a broken leg, to Hermione, who, unlike Ron, had already had no shortage of moments to shine in the films.
The Dementors are growling, hissing, flying ghouls who, while effective for what they are, are just classic horror-monsters as opposed to disturbing, otherworldly beings who didn't need to pose any physical threat, or even move at any speed, because the mental effects of their sheer presence was enough to incapacitate their enemies. The changes to their design and behavior also make them more animalistic, even though they are sentient humanoids in the books albeit ones much weirder and more threatening than Goblins or Elves.
Then there were the changes compared to the previous two films in the series:
Alfonso's complete redesigning of Hogwarts geography, as well as the general shift in tone, with more openly-comedic moments but also a general Darker and Edgier feel that contrasts from the more whimsical tone of the first two films. (This was meant to translate the introduction of darker themes in the books as the series went on, but in the books it was a progressive, creeping growth that you didn't necessarily notice as you went, whereas the transition in the films was more abrupt.) Regardless of the actual tone, the fact that the visuals were literally darker and less colorful was controversial for people who preferred the previous films' brighter and warmer color-design.
Though a recast was unavoidable due to Author Existence Failure, the radically different new Dumbledore didn't sit too well with quite a few people; for one thing, little effort was made to make him look or act like the Richard Harris portrayal of the first films, even for really basic things like his haircut or trademark gold-rimmed glasses. Indeed, regardless of his acting, Gambon's Dumbledore physically looks less like his book self than Harris's.