Final Fantasy XIII/Heartwarming

Hope's father. "You're my son! This is your home!"
- Shortly before that, the normally icy Lightning seeing Hope's distress and sweeping him up in a hug, then helping Fang scoop up Snow and saying affectionately "He's too stubborn to die,", making it clear that she cares.
  - What makes it better is if you remember that she said the same thing about him before, but more ruefully.
Lightning restoring Snow's faith that Serah can, and will, be saved.
- Speaking of Snow and Lightning, the conversation they have about what they'll do when Serah's no longer crystal. Seeing Lightning soften up is pretty poignant. Especially when, as the scene fades out, you see her rest her head on Snow's back.
The exact moment when Lightning realized that she was being an Aloof Big Sister when she realized that Snow was the only one who stood by Serah when she became an l'Cie.
XIII is really just full of them. Fang and Vanille, Lightning and Snow, Lightning and Hope, Hope and Snow, Vanille and Sazh. Really, every other cut scene is gonna have some touching dialog.
- Snow and Hope conflict. Snow's absolute refusal to leave Hope and his determination to protect him even after Hope tries to kill him to make up for the past.

Hope: You wanna die? Well, too bad. You can't. *Paradigm Shift* I won't let you!

This quote:

Snow: "Serah...my one, my only...will you marry me?"

After Vanille's Eidolon battle, where Fang tears into her about how she is the one person Vanille shouldn't have lied to, given all they've been through together. She rears back to punch Vanille and ends up just bopping her lightly before reminding her that they are family.
The ending, period.
Hope and Vanille's scene on Gran Pulse. Him joking about having feelings for Vanille, and her cute reaction...so adorable!
The scene where Snow decides help Serah fulfill her focus instead of abandoning her is all kinds of sweet. He's literally saying he'll take on the entire world just for her.
In Final Fantasy XIII Episode Zero, despite being as resentful as ever that Serah is dating Snow, while trying to figure out what she sees in him, Lightning comes to realize that she's is the one who's been alienating her sister and feels genuinely guilty for neglecting her ever since she got Promoted to Parent. She then proceeds to try to make amends, by actually planning to take vacation time off to do nothing but make up for lost time and talk to her sister, promises herself to make more time for her in the future, and comes as close as she ever has to be giddy about the whole plan. Of course, the game reveals that this goes disastrously awry, but the whole episode really brings out the much needed "heart of gold" in her personality.
In the end, Snow steals a moment away from Serah and Light, and talks about planning the wedding. He tells Light "I will make her happy". And Light just says "I know." D'aww, Light, you weren't so icy after all, were you?
Just before Snow makes his awesome entry with Fang. Lighting and Hope are completely surrounded, and Lightning tells Hope to run while she holds them off. "You survive."
Sazh's "Many things can be forgiven, but shooting kids is not one of them." It comes right after he realizes that he almost shot a very willing Vanille in retribution for his own son turning to crystal.
Snow's devotion to Serah throughout the whole game.