Given that the world isn't doing so well, you wouldn't really expect to see too many heartwarming scenes, but anything involving Jungo and the injured cat he adopts count.
The end of Io's Fate route has her talking a child out of killing himself by letting him know that he isn't the only one who lost his family and that as long as he has people who care about him, his life isn't pointless.
Yamato isn't exactly the kind of guy who inspires this, but his Fate route has two separate instances that are rather heartwarming. At Level 3, you can introduce him to tayokai as part of an earlier promise to show him what an ordinary life is like, and he winds up enjoying it. In his Level 5 rank, a dying man gives him a piece of candy he was going to give his child, telling him with his dying words he knows Yamato is with JP's yet bears him no hatred, giving said candy to him because he wants him to have it. Yamato reveals he does have a soul when he wonders if someone like him even deserves such a simple gift, revealing that on some level, he does acknowledge how cold and distant he is to others and regrets his own inability to return empathy to others.
If you're on Yamato's path, you get some mail from him right before the end of the second-to-last day. He's still repeating what he had been saying for most of the game, but what makes it a heartwarming moment is the last sentence and the picture ( an almost-finished plate of some takoyaki, his new favorite food that you helped him come to love) he's attached to the message.
Yamato: But for now, let's think about tomorrow, when you and I will change the world together.
The Nicaea avatars have moments of this that double as Foreshadowing. If you save someone via Nicaea, both the male and female Tico are sincerely grateful you used Nicaea's services to save lives. And, when you remember they were created as digital extensions of the Anguished One's will, it should be obvious what that says about him.
One of Makoto's events has her and the protagonist stumbling upon a group of rioters who are surrounded by demons. The rioters, clearly frightened for their lives and knowing that the chances of help showing up is next to none, decide to force their way past the demons despite lacking the power to properly fight back. Why? All to protect a child that is part of the group.
In Alcor's route, Yamato reveals to the group that Alcor is a Septentrione. Despite this, the protagonist just smiles and still accepts him as a friend. The heartwarming is immediately subverted because Yamato tries to kill everyone, resulting in Alcor killing him in retaliation.
Any time the hero or his party calls Al Saiduq a friend (and you have the option to call him such at least three times), and his reaction.
The culmination of Otome's Fate Route. She has a strained relationship with her adopted daughter, Koharu, to the point that the young girl runs off on her own into a demon-invested Tokyo. You go search for her and when you find her, Koharu tells you that she doubts her mother really cares for her, as she is not her "real" daughter. At that point Otome enters the screen and runs towards Koharu. The girl takes a few steps back, worried that her mother is mad at her. Otome stands still... and then rushes forward, not to berate her daughter, but to glomp her and tell her how worried she was and how much she loves her. Koharu silently stands in her arms for a few moments, not knowing what to say or do... until she hugs her mother back. As you decide to give them a moment of privacy and leave the scene, you can still see them holding each other.
It becomes even sweeter when you do the follow-up event, when you will see mother and daughter walking beside each other, singing about cute demons ("cute" according to Koharu, that is). The game specifically mentions how close and happy they look again.