France lecturing Italy about never forgetting about someone they've lost, saying that as nations they must bear with that burden. It becomes sad once you realize he might be referring to Joan of Arc.
Another moment goes to France with his pep talk for Canada. As the other nations are gathered around Italy's (supposedly) dead body, Canada sulks in the bathroom about being useless. France finds and hears him mope about being the only one unable to contribute to the situation. Being the quiet and invisible nation that everyone practically ignores, his feelings are understandable. However, France reminds him that he was the one to solve most the puzzles, and that even if he can't help with their current situation, he will always have a chance later on. He even goes so far to say that his son was better than him. This is France we're talking about! This is the man who makes suggestive comments and flaunts his fabulous naked body every chance he gets! Seeing France as a sensible person, and especially as a good father, towards Canada earned this troper's respect and definitely qualifies as a heartwarming moment.
Immediately after this, Alfred barges into the room, supposedly about to ruin the moment. Then it turns out that he was looking for Canada, as he noticed his brother was missing and got worried. Canada even notes how much things have changed, as America used to always ignore him due to how unnoticeable he was. Right after this, America tells him that without Canada around to raise his sensible opinions, he'll start doing reckless things again. The character development between the two brothers is really quite touching, though often overshadowed in favor of the relationship between America and England.
Italy and Romano's phone call.
Romano: Say my name!
Romano: No! Say my full name!
Italy: ...Romano Italy.
Romano: That's right! I'm Italy, too!
Prussia coming back for Germany. "You want to know why I came back? Because I can't think of a reason why I wouldn't help my little brother!"
America telling Italy to leave him with Canada and England's mangled bodies in one of the timelines.
"The truth is, I want to be with them until my last moment. Because they are both very important to me."
This exchange occurs when Austria arrives with a whole host of nations to rescue those trapped in the mansion.
Austria: It's rather uncanny, us all agreeing so much.
Spain: Of course! Even if we fight and don't get along, we're still fellow nations!
Russia and his sisters reuniting at long last. Doesn't help that Russia tells them to go home but they choose to stay because they are worried sick for him. And then Belarus throws him a scarf that she and Ukraine found as a reminder that he still has them.
The reason they leave the cave? They hear gunshots and realize that Hungary and Liechtenstein are fighting an enemy. Belarus is torn between helping her beloved brother and her new friends. Belarus!
The moment when Italy manages to make the jump from "I want all of you to escape" to "I want to escape with you." Being willing to die for them was exquisitely noble. Deciding that he'd rather be there for/with them proves that he not only loves them, but accepts that they love him too.
The part where Italy and America are talking after America sees England die in a past loop. America asks if there were any loops where England was the only one to die and wonders why Italy went back even then, when the only one to die was someone Italy didn't really like. Italy tells him that he doesn't really hate anyone and his goal is to get all of them out safely, because they're all fellow nations and they all have to help each other in times of need, no matter what may have happened between them in the past.
"Believe in one another. Help one another. Rely on one another. Also, escape all together."
The part where Japan overhears a conversation between England and France, where England tries to tell France - in his ridiculous, Tsundere way - to take care of America and Canada if he doesn't make it. And France basically tells him, "No, you idiot; YOU raised them with ME, and you are NOT leaving me alone with them!" Then he goes on to explain to England that it isn't just about America and Canada, or even about France himself - but that England has made many friends here, and they all need everyone to get out, including him. The idea of France comforting England, of all people - and of England going to France for help - was very moving, and spoke greatly to the game's themes of trust and friendship.