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.
The fact that Snape was entrusted with brewing Lupin's werewolf potion and making sure they're always available. He still despises Lupin for his actions in their school days, but he does his duty.
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.
There's also the fact that Dumbledore doesn't care a jot; he knows Sirius is innocent now — that is good enough for him. Besides, he's headmaster; if he says it is acceptable, then it is freaking well acceptable!
Harry's thirteenth birthday, the first time he got birthday cards and presents from his friends. The narration points out that this is the first time in his life he felt glad it was his birthday, especially since his twelfth birthday in the previous year was his worst.
Harry's two weeks of freedom in Diagon Alley is also pretty heartwarming. After a particularly horrible stay with his awful family and a near-miss with a Despair Event Horizon, he is given a very generous amount of slack from Minister Fudge and is given free reign within a capital wizarding hub town.
The last two victorious Quidditch matches against Ravenclaw and Slytherin, the Grand Finale of the original Quidditch storyline.. What's particularly heartwarming about it is the fact that they were doing pretty badly at first—-Harry lost not only his first game but ended up getting his broom destroyed. 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 realize just how loyal Peter was to his friends when he was younger and think what must have happened to make him become a traitor.
Also the fact that James, Peter and Sirius were twelve and didn't abandon Lupin like most others would.
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...
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.
Elaborating on this, Lupin lost his three best friends over the course of two days and spent the better part of thirteen years alone and friendless. Meanwhile, Sirius was locked in prison for that same period of time for a crime he didn't commit and had everyone in the wizarding world thinking he was everything he had fought against. Once Lupin enters the Shrieking Shack and Sirius confirms what happened, they embrace, each overjoyed that they have one of their best friends back.
The fact that the whole student body is sad to see Lupin go. To them he was never a monster but one of the best teachers they'd ever have.
Hell, when Harry finds out from Hagrid that Lupin resigned, he immediately runs to Lupin's office to beg him not to leave.
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!
Hagrid sticking up for Hermione with Harry and Ron. Telling them gently but firmly that they should be better than this and not let their friendship with her go. Hagrid is Harry's friend and loyal to him through and through but he's not above telling him when he's gone astray.
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.
The scene with Sirius and Harry being parted in the film is even more poignant; Sirius doesn't care in the slightest that the Wizarding world does not know he is innocent - as long as his godson, the one living reminder of his friends Lily and James, does. That's all he wants.
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, damn it, and he won't let anyone hurt his students.
What's even more awesome: just before he saw the werewolf, he was boiling with rage over Harry assaulting him. The second he sees the werewolf about to attack the kids, he goes from "what the fuck, Potter," to "get behind me, children."
While the short scene with Harry and his roommates eating those strange candies is supposed to be there for comic value, it's also heartwarming in the sense that it's one of the few times in the entire series where we see Harry look like the young boy that he still is - enjoying some pure fun with friends with all of his usual worries at the back of his mind.
Sirius on Hogwarts:
Sirius: It's beautiful, isn't it? I'll never forget the first time I walked through those doors. It'll be nice to do it again as a free man.