In Lion's Arch, you can find a human girl and a Charr cub playing "The Charr and the Ghost", running around and having fun. After they finish playing, the Charr asks if the human ever wonders why their people used to fight. The human says she doesn't care, all that matters is that the Charr is her best friend. When you talk to her, the Charr says that she and the girl are going to join the Charr's Ash Legion, and once they become tribunes, they're going to make it possible for humans to join legions. A likely impossible goal, but it is damn heartwarming to see two young kids who not only don't care about the prejudice between their people, but actively want to undo it.
Player: Times are changing.
That's not all - they're both playing as Charr, running away from imaginary ghosts.
May not be entirely impossible, given that we see Rytlock pass a Blood Legion pendant to Logan in the prequel novel and recognize the human as a brother (which for Charr, means a member of the warband rather than blood sibling) — another heartwarming moment, given how the two act throughout the book. The girl, at least, is almost a warband member and fahar sister. That the kids don't even consider that it might be a dangerous or risky thing, in a world where the two races were at war less than a decade ago, means a lot still.
In recent updates, we find that both girls were orphaned during Scarlet's attack on Lion's Arch. Luckily for the girls, an Asuran Order of Whispers agent, Brokka, and her new husband, Captain Tokk, decided to adopt them.
Brokka: They may be at a cognitive disadvantage due to their race, but they'll still benefit from our love and support.
In the same vein, if you're human and a Whispers agent, your character's friendship with Tybalt Leftpaw could be considered a Heartwarming Moment too, especially when you consider that humans and Charr in general have a hard time working together. Doubly so because it turns out that your human character showed more respect and loyalty to Tybalt than his original warband ever did.
A Norn character will be recognized by some of the Hoelbrak NPCs as an old friend. One of them will trade (bad) human pickup lines they overheard, then admit the Moot is a lot quieter without the player around.
Overheard in Metrica Province, a young female Asura progeny talking with her (often malfunctioning) golem.
Rubi: How could you deliver the wrong...? Wait, he thinks my mind is a beautiful thing?
Zojja's hero worship of Snaff gets a little creepy at times, but compared to how snarky and disrespectful she was when he was alive, the establishment of the Snaff Prize is still very touching.
During the story mode in Arah, Logan volunteers to stay behind to release the airship, getting everyone up in the air and effectively leaving him to certain death at the hands of the Risen below. Rytlock has this to say:
Rytlock: If you get yourself killed, I'll hound you into the Mists!
And when everyone finds out that Logan made it out alive, Rytlock seemed the happiest among all of them.
In the basement of the Almuten Mansion in Gendarran Fields, you'll find the Asuran child of the owners, Miss Mipp, along with a Norn Lionguard named Tholaum watching over her. If you talk to the Norn, he'll complain to you that he would've left the mansion years ago if it weren't for their child; and if asking the Norn if the progeny has charmed him, he will admit that she did. And he's pretty protective of her, too.
And when she does get kidnapped, he goes to rescue her. On the way he'd say:
Tholaun: When I left the homestead, seeking glory, protecting a child wasn't what I had in mind. That child, though, she grows on you.
And after saving her:
Miss Mipp: Psst. Tholaun. Can I tell you a secret?
Tholaun: You can tell me anything.
Miss Mipp: I was really scared, Tholaun. I tried to be a brave norn, but those pirates were really mean!
Tholaun: It's okay. You didn't cry. I'm proud of you.
Miss Mipp: Can I cry now?
Tholaun: If you insist.
In the Order of Whispers storyline, Tybalt gives Demmi Beetlestone one of his Charr rifles to defend herself with, even though it's nearly as big as she is. After he dies at Claw Island, whenever she pops up later in the game as a member of the Order, it's her default weapon. The gift really meant a lot to her.
In Bloodtide Coast, there's a fairly well-hidden Quaggan cove called Breekeelee. At the entrance there's a guard that tells you that this is a Quaggan holy site, and they keep their most precious treasures within. I swam along, expecting to see giant pearls or giant sculptures or something similar. It wasn't until I was done looking around that I noticed there were Quaggan babies swimming all around me, following me around the nursery. That's right, the Quaggans' greatest treasures are their own children. And top that off with baby Quaggans being absolutely adorable to make this one of the most heartwarming moments in the game.
In Metrica Province, there's a well-hidden cave across the lake opposite of the Funhouse with five signs that normally belong to the Inquest, all of them warning the player and demanding they turn back immediately. If you press on, inside the cave is a lone Asura named Snrif surrounded by half a dozen bunnies. Talking with her, she'll tell you she stole the Inquest warning signs and claim that she's performing dangerous experiments on the bunnies, and that it's dangerous for you to be around. If you press her and refuse to leave, she tell you she just wants to sit with her bunniesImean do her experiments in peace. When you point out her little slip, she admits that she just likes bunnies and comes to the cave to feed and take care of them, but doesn't want anyone to know because she gets weird stares. You can respond by saying there's nothing wrong with it, or be a jerk and call her a sissy for it.
The Story modes for Citadel of Flame and Crucible of Eternity. Throughout the former, Logan and Rytlock start out with Teeth-Clenched Teamwork, but gradually get less vehement with each other. By the end, they finally agree to start patching things up. In the latter, they're shown to be much more friendly and eager to work together. It culminates at the end when they try to encourage Zojja to rejoin Destiny's Edge.
Logan: Zojja, we all miss your mentor, but I think it's time to let things heal. Come back to the group.
Zojja: Come back? You mean work with Eir Stegalkin? Never. I can't forgive her.
Rytlock: I forgave Logan.
In the Charr story, if you selected the Honorless Gladium as your sire, you meet your sire's Norn friend, who immediately recognizes you as his friend's cub. Since the character is customizable, but the sire's appearance is constant, it's very unlikely that your character bears a lot of resemblance to their sire...which means that he described you so well and talks about you enough for his friend to know exactly what you look and/or act like. Vallus Smokemane must be really proud of his cub!
Completing your personal story gets you a rousing hero's welcome at Fort Trinity. Everyone is absolutely ecstatic for their victory over Zhaitan. Everywhere you go, people of all races salute and cheer for you, and those who aren't are busy simply celebrating with their friends and comrades. The bonds between Destiny's Edge have been reforged and they're already gearing up to take down the next Elder Dragon. And after the final cutscene plays, you get to listen to "Fear Not This Night"!
During the Escape From Lion's Arch event, players poured into the city on an hourly cycle to join in quests all over the map that involved aiding and evacuating survivors of Scarlet's attack, and keeping control of rallying positions within the city. No quest saw more regular participation than the one where a handful of Lionguard need help leading a group of children to safety. Every Aetherblade Pirate that so much as showed its face anywhere nearby would get hit from twenty directions by the absolute mob of players providing escort.
The May 19th update introduced a severe bug in World vs. World that prevented players from targeting their opponents, effectively breaking combat. Rather than wait for it to be fixed, the Gunnars Hold, Riverside, and Elona Reach servers (which were scheduled to be fighting each other) took the opportunity to celebrate this unforeseen instance of "world peace" and held a peace march together across the Mists. Doubles as a Crowning Moment of Awesome as another example of the community doing awesome things in spite of setbacks.
During the Living Story Season 2, Taimi the Asura has a moment where she takes her newly created calibration device - which she is very proud of - and tries to run with it away from Councillor Phlunt, who wants to take control of it for the Synergetic college of Rata Sum until Taimi is of age. She is ambushed by the Inquest on her escape, and her golem is disabled, leaving her helpless and terrified: a tone you never hear her express genuinely till now. The Player, Braham and Phlunt's party go after her and manage to protect her from the Inquest. The Heartwarming Moment is seeing that Braham doesn't once chide Taimi for acting so recklessly, instead being gentle and caring as he tries to calm her down as she is sobbing in terror then decides to stay with her till she is ready to go.
Heart of Thorns Moment: During one of the last missions, you come across Zojja and Logan trapped in Mordemoth's life pods, sapping their life away to fuel its army. Once Zojja is freed her first action on seeing you is to ask where Taimi is, as neither she nor Scruffy are with you, and she doesn't stop until assured that Taimi is ok.
One bright spot in the chaos of Living Story Season 3: Garm survived the Pact Fleet's destruction! Rox brings him home to Stonewright's Steading to recover during Eir's memorial, something everyone acknowledges as the best possible farewell present.
If you're playing through Heart of Thorns as a Sylvari commander, Canach has some unique dialogue with you. Some of it is flirty, but a lot of it involves him opening up to the Commander in ways he doesn't do if the Commander is a different race. After everything that sad cactus has gone through, seeing him finally start to make friends with someone is nice.
Another sweet thing about this is the fact that near the end of the battle, the Sylvari Commander starts to refer to Canach (and Treahearne as well) as 'brother', cementing that they truly became very close due to the Modremoth incident.
Caithe / Canach helping the Sylvari Commander break free from under Modremoth's control is as heart warming as it gets. They both believed that the Commander was stronger than the influence, but when Modremoth proves too powerful for them, they lead the player and manages to help them break free from it's control. Canach wishes to repay the Sylvari Commander for the trust and compassion they had in him while Caithe wants to help the player, regardless if they had lost trust in her.
The February 22, 2017 Balance Patch included a nice current event. It involves a send-off to Trahearne who died at the end of Heart of Thorns storyline. Not only do you get to repair Caladbolg and be blessed as its new wielder as a way to finish off that plot thread, you get to talk to Trahearne, closing up his fate in a more heartwarming way as he talks to the Commander one last time. Many people consider this current event a really well done one.