The scene with Raz's father before the final boss battle:
Raz: How can you hate me for being psychic, if you're psychic too?
Raz's Father: Son, I could never hate you. I only wanted you to be happy, Razputin. And safe.
While some of it is played for laughs, the speech Raz gives Oleander to convince him to stay in the beginning cutscene can count. After it's done, the coach who was just finishing lecturing the other campers and talking about how they were going to die if they failed their training was on the verge of tears.
Raz: "But in this dojo... In this psychic dojo... They make you a hero."
It's very probable that Oleander wrote the pamphlet speech that Raz recites out of his own experience. If so, not only does Oleander realize that Raz was discriminated against just like him, but also that Raz sees Oleander as the hero who gave him hope. Given Oleander's backstory, this is huge.
You have to go idle for a while, but watching Raz slowly start to dance in Milla's Dance Party is just so adorable.
Using Clairvoyance on the staff can yield heartwarming results. Milla sees Raz as a sweet little baby to look after owing to her own tragic past. Raz reminds Sasha of himself as a youth so much that he sees Raz as a younger version of himself. Ford Cruller sees Raz dressed up in a full Psychonaut uniform — as far as he's concerned, Raz is already worthy of the title.
One might notice Ford sees Raz as Sasha Nein, only younger. Keep in mind, Ford knows all about Clairvoyance, and it could be he's projecting that image to give Raz confidence - it doesn't make it any less heartwarming.
While Raz is collecting all the brains of the campers he gives each one a kiss. Even Bobby Zilch. It's kinda cute.
Raz has a funny remark to say about every brain he retrieves, but when he finds Dogen's? All he says is this:
Raz: Dogen Boole. I told you I'd get your brain back.
Raz saying "What a magical lady!" in reference to Linda.
Technically not part of the game itself, but in the official Vault Viewer commentary for iOS, it's oddly sweet to hear Scott Campbell (the official artist) refer to Tim Schafer, his boss, as "bro".
When seeing some of the other campers making friendship bracelets, Lili calls the idea stupid. Raz remarks to her that he wouldn't mind getting one, to which she responds by getting flustered and calling him stupid. Later in the game, we see that Lili has made him a friendship bracelet. Aww...
Fred's relationship with the other inmates post-cure. He's genuinely laid back and cool (and warm towards his former charges, being the orderly.)
Helping the inmates regain their sanity, and leave the Asylum for the first time in years.
The way Raz treats the inmates once he's in their heads. From trying to coax Bonita out of stage fright by telling her that, as an ex-carny, he knows how she feels, to helping Edgar let go of his past, to being undyingly optimistic and encouraging during the Bonaparte battle... you gotta hand it to the kid; he's a bit cheeky, but his heart is definitely in the right place.
Sasha Nein seeing the loving memories his distant father still holds for his mother who died years before, even if he was also unfortunate enough to catch the tail end of a sexual memory as well.
Milla's Ship Tease moment with Sasha. The biggest smile Milla gives in the game is when Sasha's smiling back at her.
When you consider the fact that in this world, psychic powers go hand in hand with mental illness, it's heartwarming to see most of the characters portrayed as average, likable people, all of whom Raz treats with determination and empathy.