Daphne's fantasy sequence as she lays dying. "Fly me to the moon". Oh god...the only way it could be more heart wrenching is if someone made an AMV of it.
Hiro saying goodbye to Kaito after visiting his mother's funeral.
Hiro's magic show/speech to Emma in Tabula Rasa. He teaches her that powers are always a blessing if you use them right, while putting on a magic show for sick kids in a hospital. (Not to mention, this is also during the Cast from Hit Points time, so he himself is dying the entire time.)
There's the Bennet family hug at the end of "Company Man" - really heartwarming given how much has happened to their family in that episode. They got taken hostage, Claire was shot, and to cap it off their house got nuked by Ted Sprague.
And Molly Walker has to be a walking Heartwarming Moment. Starting with that one look she gave to Micah in the season 1 finale.
Niki has been put through a lot of stress in the first season, so it's especially heartwarming when she finally overcomes her Superpowered Evil Side, punches out a Company agent who's been masquerading as her, and saves Micah by herself in the season finale.
D.L. phasing his hand through the glass when he and Micah were visiting Niki in prison was definitely a heartwarming moment since Niki turned herself in so that Micah will at least have one parent.
Matt at Molly's hospital bed in the season 2 episode "Out of Time."
Matt: I love you, Molly. Molly: I heard!
This troper liked the honest last few words between Peter and Nathan at the end of the Season 1 finale, considering all Nathan's magnificent bastardness beforehand.
For this troper, it was in season two: Peter has lost his memory, and ended up in the future with no idea how he got there. The world is in a crisis, and his mom is called in to help him. She tries to get him to remember, and finally, he does, accompanied by a montage of everything nice she'd ever done for him.
It freaked me out at the time, because we didn't know what her power was and I assumed they were fake memories. I don't know why, I just did. I know better now, though.
This troper finds three Claude-to-Bennet moments in "Company Man" strangely heartwarming: "By 'them' do you mean people like me?", "This isn't the first time you've been asked to kill a man, but is it the first time you've killed a friend.", and "Evidently, I think you're a better man than they do."
Season 3: "Claire Bear, huh?" and Hiro's last meeting with his mom.
The Paris scene from "Cold Snap": Matt gives Daphne a storybook ending; he telepathically creates a dream in which he flies her around the Eiffel Tower then to the moon (per her request) while she lay dying in a hospital bed. Perhaps the most heartbreaking scene of the entire series. Definitely the most romantic scene. It's yet another CMoA for Bryan Fuller, for that matter.
Also in that episode, Tracy's sacrifice right before the scene above. Micah snaps at her for giving him up to the government. Tracy responds by telling him to escape, then sacrificing herself to kill the agents so that he can get away. The blink and the tear at the very end as she's lying on the ground really adds to the effect.
Parkman's reaction to meeting his baby, which made this troper like him for the first time.
Peter and Emma playing the piano together, especially in "Shadowboxing", after the talk they had about her nephew's death and her saving the little girl's life. He then put the girl's tiara on Emma's head. D'awww.
"You've found your way home, brother."
Nathan convincing Peter to let go of him and accept his death.
Nathan: "You can do anything, Pete, anything.
Going back to Season 1, there's a moment of the roof where Peter and Claude are hanging out with Claude's pigeons and Peter asks what Charles Darwin meant by "a species' full potential." It's all in the way Claude looks at him:
Claude: I think he meant you, friend.
Peter and Claire share a few familial moments together after Nathan's funeral.
Hiro and Charlie's reunion doubles as a Heartwarming Moment and a Tear Jerker. While Hiro recovers from his brain surgery, he receives a note from Charlie, who is apparently somewhere in the hospital. Hiro rushes to her room to find that Charlie is now an old woman. Samuel had her put in 1944, where she has lived a full and happy life to this day. She has a husband, four children, and seven grandchildren, and she wanted to see Hiro one last time before she died. Hiro's quiet bow goodbye and Charlie's happy face make this a Heartwarming Moment for the ages.
Pretty much all of Sylar and Peter's interactions in "The Wall" and "Brave New World". (It also doubles as fuel for any Sylar/Peter shippers.)
Sylar and Peter are trapped in Sylar's head for five years (a few hours in reality) in "The Wall". Eventually, the have to get along in other to escape. Sylar apologizes for killing Nathan and Peter seems to forgive him. Then Sylar says this:
Sylar:I might not be able to bring Nathan back, but I can sure as hell swing a sledgehammer. (Sylar then helps Peter dismantle the wall and they are able to wake up.)
After they both wake up, Sylar asks if the time they spend together was actually real and Peter only offers a small smile.
During "Brave New World" Peter and Sylar split up to find Samuel and save Emma. Sylar actually gives Peter a reassuring pat on the chest by putting his hand over Peter's heart. Awww.
Peter and Emma's reunion at the carnival. The happy look on Peter's face is adorable.
Sylar greeting Baby Matt Parkman. Sure, he's just pretending to be nice and polite to Janice and Baby Matt in order to keep Parkman from freaking out, but it's adorable nonetheless.