"I dare to dream that she might even care for me..."
When Esmeralda frees Quasimodo after he being tied down and humiliated by the crowd. Also counts as a Crowning Moment of Awesome, as she was defying Frollo by doing it AND (for that time) got away with it.
Just before that, when some of the people are attacking Quasi. Phoebus looks at the attackers disgusted and asks Frollo if he should step in and put a stop to it.
Towards the end, when Quasimodo walks out of the cathedral, expecting to be ridiculed and shunned like he has been all his life. Then a little girl approaches him, touches his face, and gives him a hug. Say it with me: D'awww.
The moment before that is very sweet, too; Esmeralda wordlessly beckoning Quasi out into the public, all with the warmest smile that seems to say that things will be alright to Quasi (and from the layout placing, the audience).
To make it even better, there's some foreshadowing related to this scene. There's a scene in the movie earlier where Quasidmodo is talking to his gargoyle friends, angrily asking "What am I supposed to do? Go out there, save the girl from the jaws of death and have the town praise me like I'm some sort of hero?!" Everything said in that sentence comes true in the final act of the movie.
Quasi blessing Esmeralda and Phoebus' relationship. Despite most saying that this breaks the movie's "looks aren't everything" message, this shows a lot of growth in his character in being able to let Esmeralda love someone else and make peace with Phoebus. Extra bonus for the melody from "Heaven's Light" playing.
Arguably it is not broken as the characters are loved for their personalities. Phoebus is shown to have inner beauty himself and slight character development from a reluctant pawn of Frollo to someone who disobeys orders to rescue an innocent family. He is attracted to Esmeralda at first sight but loves her for spirit and energy, and she loves him because he is kind at heart. To add to the Heartwarming list, Phoebus completely credits Quasimodo for finding the Court of Miracles when he notices how Quasi looks forlorn at being ignored.
It's even more heartwarming when you realize it does, in fact, keep with the movie's real Aesop: the comparison between Quasimodo and Frollo. By blessing Esmeralda's relationship with Phoebus, Quasi proved that he's the real man despite looking like the monster. He's happy knowing Esmeralda is happy.
"God Help The Outcasts"
When the little girl brings Quasi into the crowd, and Clopin calls "Three cheers for Quasimodo!" And he's carried out into the crowd. He's happy, he has his freedom as he desperately wanted, he didn't need a girlfriend to get his happy ending.
Quasi's interaction with the baby bird at the beginning after the prologue. According to the commentary, the makers wanted a more humane Quasimodo, as opposed to the monstrous ones of past adaptions. They got it. When the gargoyles are trying to convince Quasi to take part in the Feast of Fools instead of watching, they even use his earlier "No one wants to be cooped up here forever" line he used to encourage the bird.
Knowing the story a bit can help with the bird scene, as his line "No one wants to be cooped up here forever" clearly refers to himself moreso than the bird.
Right after Frollo's fall into the flames and chaos, it seems that Quasimodo will follow the same fate when Esmeralda looses her grip on him, but luckily Phoebus grabs him in time and saves him. How does Quasi thank him? With a smile, and then a hug. Cue Phoebus patting Quasi's back, and smiling as well.
Quasimodo's mother; she literally died trying to save her son (it's nice to know that Quasi's biological parents, especially his mom, still loved him despite his appearance).
The way Clopin defuses the situation after Quasimodo is identified by a villager at the Feast of Fools is heartwarming in its own way.
The scene where Phoebus first talks to Quasimodo right after helping Esmeralda escape. Quasimodo is already pissed that a soldier is in his bell tower and tries to be intimidating in order to drive him out of the church demanding him to leave, waving a torch wildly at him. Phoebus, though scared manages to explain why he wanted to talk to her which causes Quasimodo to stop.
Phoebus: Tell her I didn't mean to trap her here, but it was the only way I could save her life. Quasimodo: I'll tell her if you go. Phoebus: Oh, and one more thing: tell Esmeralda - she's very lucky. Quasimodo: Why? Phoebus: To have a friend like you.
The Disney sequel:
For all of its flaws the sequel does have its own genuinely heartwarming moments, like Quasimodo's relationship with Zephyr and Madellaine admitting her love for him.
Bonus points for the Meaningful Echo Madellaine uses to convince Quasimodo that she's on his side and wants to help: "There's more to me...", a phrase that's as true about him as it is about her.