Teen Titans (animation)/Heartwarming

Comics

There was a single issue dealing with Tim Drake after his father had been killed in Identity Crisis. Taking the Batman route, Robin shuts his feelings in and tries to act cheerful (ending up being more cheerful than normally), and only Conner really knows what's bothering him and encourages him to talk to the team about it, which Tim still refuses to do. Later, when they are chasing Electrocutioner, said villain begins making father-insults to Tim (mainly alluding Batman as Robin's father, since he does not know the boy's identity), which causes Robin to beat him to a pulp in a sudden rage before the other Titans stop him. When he finally tells them what's wrong and Wonder Girl asks if he wants to talk about it, he hesitates before looking at them with tears in his eyes, muttering "Yes. Yes, I think I need to." We then close the issue with a panel of Robin being hugged by Wonder Girl, with all the Titans there for him and these words:

Robin: For the first time in a week, I stop thinking like Batman. And I start thinking like a Titan.

During the Dark Side arc of Teen Titans, Kid Devil goes into an Unstoppable Rage and almost kills a brainwashed Hardrock, venting out out a great deal of pain and frustration from being seen as the weakest link of the team, beaten up, tortured by Clock King and told that no one loves him in order to condition him into a monster for fighting in the Dark Side Club. Miss Martian gives Eddie an "I Know You're in There Somewhere" Fight speech with her telepathy to bring him to his senses.

M'gann: Eddie. No. Eddie, Please Listen...This isn't you. You're not a killer. What they did to you--what Clock King said to you... None of it's true. You're not a monster. You're not alone. You're not unloved. You're a sweet boy who had a dream to be a superhero sidekick and got to live that dream. You're thoughtful, funny and earnest. You're a Teen Titan, Eddie. You're a good guy.

This scene, where the original five Titans — Dick Grayson (Nightwing), Donna Troy, Garth (Tempest), Roy Harper (Red Arrow) and Wally West (The Flash III) — run across statues of them from way back when they were teenagers — as Robin, Wonder Girl, Aqualad, Speedy, and Kid Flash. They comment that even though their lives have taken so many twists and turns since the '60s that it's a miracle they're all alive, they are still together, still friends, and still Titans.
Basically any time Roy Harper is together with his daughter Lian it is bound to make a reader go "aww" or start crying.
Teen Titans: Year One, the modern look on how the original five founded the team, ends with Batman at his computer, trying to think of something that will convince Dick to come back and continue to be his sidekick. After multiple attempts with lengthy, emotional sentences, he settles on three simple words. "Robin, come home".

Animated series

During the battle with Trigon, Robin is carrying the 10 year old form of Raven on his back, reminding her of why they are friends and how important she is to the team and to him.

Robin: Yeah. It's the end of the world. But so what? We're still here. Still fighting. Still friends.
Raven: Look at me, Robin! There's nothing I can do! There isn't any hope!
Robin: Then I guess I'll just have to have enough hope for the both of us.

Followed when Raven, now her right age and white outfit, takes down Trigon.

Starfire: Raven, that was...
Cyborg: Unbelievable.
Raven: No. It wasn't. (Robin comes up to Raven, and she hugs him tightly.)
Raven: Somebody believed. (Robin puts a comforting hand on Raven)
Robin: Welcome back.

Don't forget at the end of "The End" where Robin tells Raven that her battle against her own demons and her destiny has always inspired him.
Beast Boy demonstrates a surprising amount of wisdom in an encouraging speech to Cyborg in "Overdrive", after the former has worn out the chip he implanted in his brain and is discouraged about failing to catch a criminal.
The final scene of "Sisters", when Robin and Starfire are sitting watching the sunrise together and Robin tells her that, while her sister was interesting, "No one could ever take your place." really does it for this troper. She blames it on the pretty lighting.
And of course, the scene towards the end of the movie where Robin admits to Starfire that they can be more than just teammates followed by their first real kiss was every shipper's dream come to life.

Cyborg: Well it's about time!

- Can't forget this exchange at the end of the movie.

Cyborg: Guess you're not a wanted man anymore.
Robin: (Holding hands with Starfire) Oh, I wouldn't say that.

- Really, the two are a goldmine for these. Check out the entire last half of Robin and Starfire's cave conversation in "Stranded", as well as when they hold hands and Robin says "As long as we're together, we'll be okay." Aww...
From "Spellbound":

Beast Boy: You think you're alone Raven... but you're not.

- And to top it off: in response, Raven opens her bedroom door...and gives him a great, big hug (which is so unusual and OOC for her that Beast Boy is actually rendered speechless).
The end of "Aftershock, Part 2", when the dust settles after Terra's Heroic Sacrifice and life returns to normal, the Titans are seen carrying a plaque and flowers to the statue that was once their friend. It's hard to tell which part of the sequence tugs at the heart more; that or Beast Boy's speech, which mirrors and subverts the evil speech Terra gave at the start of the episode:

Beast Boy: "Her name was Terra. She was gifted with tremendous power, and cursed with it as well. She was a dangerous enemy, and a good friend. And she was one of the bravest people I have ever known."

- Or when the team has made their promise to bring her back, and we see what the plaque reads:

Terra: A Teen Titan. A True Friend.

In "Fractured," Johnny Rancid gets Larry's powers and begins unleashing havoc. Robin doesn't even try to stop him because he worries he'll mess up again and make things worse. Larry, however, tells him this:

"I mess up all the time, but I still try. That's how come you're my hero, Robin. Because no matter what, you always try."

In "Apprentice", after Slade has blackmailed Robin into working for him:

Slade: "You'll grow to like this. I may even become like a father to you."
Robin: "I already have a father." *images of dozens of bats appear in the scene dissolve*

At the end of "Apprentice", when Robin admits to Starfire that he and Slade really are very similar. But there's one big difference between them. "Slade doesn't have any friends." Then, Robin smiles at Starfire before looking over at the rest of the Titans, who are being their usual, lovable selves.
The end of "Lightspeed": As random as it might have been, the writers manage to make a lot of Kid Flash and Jinx's interactions pretty cute, starting from how when he first takes the Hive Five by surprise during a bank robbery, KF pulls pranks on all of the other Hive Five except Jinx, whom he gives a red rose. After Jinx's High Heel Face Turn, she has a rather unsure expression, but she stops in her tracks and finds another rose. After briefly spotting Kid Flash overlooking her and dashing away, Jinx walks ahead with a sweet smile.
During the origin episode "Go!", Cyborg comforts Raven ,who feels alienated after Beast Boy gets creeped out by her powers:

Raven: I don't exactly fit in.
Cyborg: He's green, half of me is metal, and [Starfire]'s from space. You fit in just fine.

"Car Trouble" is a good early one for Raven. She makes it plenty clear that to her "it's just a car,", but partway into the episode reveals that she does understand how Cyborg would be attached to his car the way he was, and at the end when rebuilding it, she actually is shown helping Cyborg with the biggest smile she's likely ever shown on the show.
At the end of "Birthmark," the Titans throw a redo-birthday party for Raven:

Beast Boy: You may not like your birthday, but we're all glad you were born.