Heartwarming: Harry Potter and the Prisoner of Azkaban
From the Book:
In Prisoner of Azkaban, when Harry says that of course he wants to move in with Sirius, and Sirius beams.
...for a moment, he was recognizable as the man who had laughed at Harry's parents' wedding.
In that same book, after they have rescued Sirius from the Ministry of Magic, he says goodbye to Harry and flies away on Buckbeak.
Sirius: You are...truly your father's son, Harry.
Also from the same book, Harry had believed that he had seen his father that night, when he had actually seen himself. He told Dumbledore that it was stupid of him. But then Dumbledore replied that his father did appear that night.
Dumbledore: Your father is alive in you, Harry, and shows himself most plainly when you have need of him. How else could you produce that particular Patronus? Prongs rode again last night.
Also, Fred and George making Harry feel better with some expertly-placed humor when Malfoy is taunting him about fainting around the Dementors.
In The Prisoner of Azkaban, Sirius mailing Harry a Firebolt and signing his permission slip to go to Hogsmeade, which is accepted as even though Sirius is a wanted criminal, he's still Harry's godfather.
The part in Prisoner of Azkaban when Gryffindor wins the Quidditch Cup. Why was this not mentioned yet? The way everyone is cheering for Harry, and Wood crying. If that isn't Heartwarming to you, you're asociopath. What's particularly heartwarming is the fact that they were doing pretty badly before then, and they would only win if they beat Slytherin by a certain number of points. Wood was half convinced that he would leave Hogwarts with no Quidditch Cup. And yet... they did it...
The whole backstory of how the Marauders became Animagi. They learn that their friend is a werewolf and, instead of abandoning and shunning him like the rest of the wizarding community would, they spend three years learning a particularly difficult piece of magic for the sole purpose of helping Remus during the full moon. James and Sirius may have been bullies, but they would do anything to support a friend in need. This also causes a bit of a tearjerker, when you realise just how loyal Peter was to his friends when he was younger and think what must have happened to make him become a traitor.
A subtle one: Sirius has survived for 12 years in Azkaban prison (an unheard of length for a prisoner to last) solely due to the knowledge that he didn't kill James and Lily. He escapes when he learns that Harry is in trouble, and lives a hellish life out in the woods for almost a year while he seeks desperately to find a way to protect his god-son. And upon seeing each other for the first time in 12 years, Harry immediately accuses Sirius of murdering James and Lily, even going so far as to attack him head on. The fact that Sirius never raises a hand to protect himself and never even gets upset at Harry for the misunderstanding...sniff.
Lupin sticking up for Neville when Snape insults him before the boggart lesson. Pretty much all of the teachers think of Neville as useless and tend to get exasperated with him, but Lupin notices his lack of confidence and tries to help him get over his fear of Snape.
The speed and ease with which Lupin accepts Sirius's innocence suggests that, even after all this time and despite the seemingly overwhelming evidence, he still harboured some doubts about his friend's guilt.
Crossing over with CMOA:
Peter: The Dark Lord... you have no idea the weapons he possesses... he would have killed me, Sirius!
Sirius: THEN YOU SHOULD HAVE DIED! Died rather than betray your friends, as we would have done for you!
From the Film:
The scene in The Prisoner of Azkaban where Lupin tells Harry about his parents, combined with the soft music in the background, did it for me. The music in question
Also, Harry and Sirius's goodbye at the end.
Sirius: It's cruel that I got to spend so much time with James and Lily and you so little. But know this, the ones that love us never really leave us. And you can always find them in here. (places his hand over Harry's heart)
Harry: But, you're innocent"
Sirius "And you know it! For now, that'll do."
The music really upped the heartwarming tear-jerker aspect of those scenes. Even better when you realize that the theme is titled "A Window to the Past," which is so fitting for this book/movie and its context to the rest of the films.
In order for the Patronus spell to work, one must think of a strong, happy memory. After a failed attempt, Harry mentioned that there is one memory that might be strong enough. He tried it and it managed to stop the Boggart Dementor. He later told Lupin that the memory was of his parents. Harry didn't even know if it was real or not, but it worked.
In Prisoner of Azkaban, a very quick moment, but one that also doubles as a CMOA, is when Harry, Hermione, and Ron are being threatened by werewolf-Lupin, and Snape's first, knee-jerk reaction is to get between the kids and the werewolf and throw out his arms in a "oh-no-you-don't" gesture. Even better because this wasn't a "oh, meh, I gotta go save Potter's sorry life again" moment; it was a knee-jerk reaction on Snape's part.
He even does it twice: When he and the trio all fall to the ground after Lupin tries to attack them, he immediately jumps to his feet and throws out his arms again. The implication in that scene is that, while Snape may have his Jerkass moments and personally can't stand Harry and his friends, he's still a teacher, damnit, and he won't let anyone hurt his students.