The end of the film, where Matt decides to honor Andrew's wishes by going to Tibet, and confirming that he really did care deeply for Andrew despite seeming to grow apart from him in high school.
Andrew comforting his mother's laboured breathing by gently rolling her over and draping the blanket over her; if you listen carefully, you can actually hear her whispering "thank you." Of course, things go south about two minutes later.
The scene with the three boys lying together and just talking after they fly for the first time, with all three of them agreeing that this has been the best day of their lives.
When Andrew's getting ready for the talent show and talking about how Steve loaned him some clothes, his mother comments how good it is that he's spending time with his friends; the fact that she goes out of her way to encourage and nurture her son even when she's virtually at death's door just makes the scene that much more heartwarming.
The way Matt refuses to believe that Andrew is a bad person. Even when Andrew goes on a Roaring Rampage of Revenge and destroys pretty much half of Seattle, Matt continues to plead with Andrew, convincing him that he is a good guy. Matt even seems genuine when he apologizes for neglecting Andrew earlier on in the film.
Matt and Steve's speeches to Andrew's camera, showing that they genuinely are his friends. As with everything else, things go south after this, but it's still a heartwarming set of moments.
Particularly Matt's video to Andrew at the party. Matt wonders when things got reversed, when Andrew became the popular guy upstairs with a hot girl, and Matt became the one alone at the party talking to a camera. But Matt doesn't begrudge Andrew for that, and celebrates Andrew's growth. Matt even tells Andrew that he loves him. In retrospect, it makes the ending more tragic, but in the moment, it is heartwarming.
Minor, but important. When Steve, the first time we see him, wants to ask Andrew if he can borrow his camera, he instantly asks "are you okay?" when he notices that Andrew seems to be crying. This guy, probably a bit drunk, on his way to get a camera from another guy to basically "film some cool shit we found" immediately asks the guy he's never really talked to if he's alright. That says a lot and really cements his Nice Guy status. Made even sweeter when he says shortly after that he actually does remember Andrew from freshman year, if for nothing else than always wearing that hoodie.