In "In the Dark" when Angel finally gets the Gem of Amarra: the look on his face as he stands in the sunlight for the first time in over two hundred years. Doubles as Tear Jerker even without the knowledge that he'll destroy the stone.
In "I Will Remember You", Buffy and Angel's one perfect night together. It obviously couldn't last, but did they have to push the Reset Button?
How about Angel giving up his mortality because he knows Buffy needs him as he is?
Oracle: He is not a lower being.
"Expecting" is littered with moments like these, proving just how much Angel and Wesley care about Cordelia. Particularly Angel's "No... I'm family," and the last scene:
Cordelia: I learned something, too. I learned, um, men are evil? Oh wait, I knew that. I learned that LA is full of self-serving phonies. No, had that one down, too. Ahh...sex is bad?
Angel: We all knew that.
Cordelia: Okay. I learned that I have two people I trust absolutely with my life. And that part's new.
Wesley: (starts to tear up and wipes his eyes) Allergies.
In "Five By Five", when Angel and Faith crash into the street in the middle of their fight, rain begins to fall and the cynic taunts from Faith start turning into desperate screams, begging Angel to kill her, countered by soft refusals by Angel, and Faith's final complete breakdown.
And later, he tenderly tucks her into his bed, putting an extra blanket over her for warmth.
Angel's entire relationship with Faith in those two episodes and later in Season 4 is generally very heartwarming.
The end of the second season premiere when Angel visits Faith in prison and he asks how she's doing and she teases him on having to sing Barry Manilow at karaoke!
Angel buying Cordelia new, expensive clothes to earn her forgiveness at the end of Disharmony. It's also a Crowning Moment of Funny (mostly for Wes's reaction) but simply seeing them jumping around with those big grins on their faces... Aww.
A double Heartwarming Moment with a bit of fridge logic. She is the only one of the team Angel is truly dedicated to winning over. Because he loves her.
Cordelia helping Angel finally deal with Buffy's (second) death toward the end of "Heartthrob."
Cordy: You're a living, breathing... - well, living, anyway... (coaxes a smile from Angel) - good guy, who's still fighting and trying to help people. And that's not betraying her, that's honoring her.
Angel: You think?
Cordy: I'm Cordelia. I don't think. I know. (smiles widely and tenderly) Okay?
The end of "Hell Bound":
Spike: I'm not going to be like him, desperately keeping myself out of hell no matter who it hurts.
Fred: Just proves what I've been telling everyone.
Spike: That I'm a handsome devil who livens up the place?
Fred: That you're worth saving.
The clincher for this: despite Spike desperately wanting to be corporeal again, he's genuinely happy for Fred doing her best to help him, and when her life is threatened he immediately gives up his chance to become corporeal again in order to save her from Parvayne, going on to say he wouldn't change his choice for the world.
The scene in "Lullaby" where Darla confesses that she truly, truly loves her baby and that she doesn't want to let it go because it gives her the ability to care. The moment when she staked herself was also a HeartwarmingTear Jerker.
Seeing Angel as a father just about did me in. How he wouldn't let anyone hold Connor after he was born and was so fiercely protective. Singing an Irish lullaby to his son. When he tells Cordelia that if he needed to see a doctor on the sunniest day of the year at noon, he would get him there. I stopped watching the DVDs halfway through season three because I knew his son was taken from him, and I can't stand to watch it.
The Angel-and-Baby-Connor moment that breaks this troper is at the end of the S3 episode Couplet, after Angel's gone through all this trouble to help Cordy and Groo live happily ever after, even though it's killing him inside. He goes upstairs, picks up his infant son and radiates pure joy. He carries the baby downstairs, where they bump into Wesley.
Wesley: You startled me.
Angel: Oh. We didn't mean to.
Wesley: I thought I was alone.
Angel: Yeah. (smiles at Connor) So did I.
Pan down to a certain prophecy...
A small moment in Inside Out on the subject of Lilah's death
Wesley: You were mortal enemies. Why should you care what happened to her?
Angel: Because you did.
A real life version comes from the special Lorne comic made in tribute to the late Andy Hallet. First Lorne sacrifices himself to save the day and the gang saying goodbye afterwards is clearly directed at Andy. Next is a poem summarizing Lorne's role in after the fall and then a eulogy by Mark Lutz who was best friends with Andy. They then explain that they got rid of Lorne in a way that they could bring him back later but so he was out of action for now so he could enjoy some peace for a little while.
I'm appalled that Doyle's brief video cameo in "Birthday" doesn't get a mention here. The expression on Cordelia's face when she sees the video and hears that Doyle never meant to give her the visions, and that they were accidentally transferred because Doyle was in love with her takes the cake for me. Also, Cordelia becoming half-demon at the end of that episode so she can keep the visions and help Angel, even though she could happily live life as a sitcom star also counts.
In "Benediction" when Angel and Connor are play fighting - an honest father/son bonding moment.
The way Dennis the ghost looks after Cordy is incredibly heartwarming.
A villainous moment was when Dru poured her heart out to Darla about why she vamped her in "Reunion".
For you. All for you. I thought it was what you wanted - to be saved. All alone. All alone in the dirt. We've lost our way and the little worm won't dance if he's told to. (Starts to sob) No. No.. (They hug then Darla kills a guy)
Wesley and Faith in 4x13, "Salvage", discussing the recent release of Angelus.
Faith: I'm not gonna kill him, Wesley. (off Wesley's expression) Angelus. Don't care what you thought you sprung me for. Angel's the only one in my life who's never given up on me, there's no way I'm givin' up on-
Wesley: I know. That's why it had to be you.
The last scene of "Parting Gifts". One word: "Breakfast?"
The very end of "Life of the Party," when Angel takes care of the extremely sleep-deprived Lorne.
He's being positively...fatherly.
Angel apologizing to Nina at the end of "Smile Time".
Angel: So...how about breakfast?
Nina: ...What do puppets eat?
Angel: (extends his hand) Let's find out.
Cordy telling Angel to smile more often in "Dead End".
When Connor acquires a leather jacket in "A New World," there are some shots that make him look just like his father.
When Cordy sticks her tongue out at Angel, telling him she can't be fired because she's Vision Girl, we see Angel struggling not to smile.
Another HeartwarmingTear Jerker is at the end of the fifth season episode "Lineage," when Wesley calls his dad "just to talk" right after killing a cyborg disguised as his father.
Before that, when Wesley killed the cyborg to protect Fred, she asked how he had known it wasn't really his father. Wesley responded that he didn't.
Also in the Grand Finale, while everyone is trying to enjoy what they expect to be their last days alive, though each of their stories are touching in their own ways, the best is probably Spike's: he reads some poetry he wrote at a club poetry slam. The way he reads it, and the earlier interaction between Spike and the bartender, suggest that he plans for this to lead into one last bar-room brawl ... But the audience cheers his work enthusiastically, meaning William the Bloody-Awful-At-Poetry is finally getting some respect. Sure, they might all have been roaring drunk, but the stunned joy on Spike's face and in his voice as he thanks them for their applause and gets ready to recite another poem is beautiful.
For extra heartwarming, compare the scene to the one in "Fool For Love". The latter has William chickening out and others turning it into a source of humiliation, while the former has him putting his work out there with no regrets.
A smaller one from the finale: Gunn visits Anne, who's still working her ass off trying to keep her shelter going. He asks her what she'd do if she had absolute proof that the whole world is rigged, that the bad guys will always win, and that nothing they can do matters. She looks at him as if it was the dumbest thing she ever heard and sums up the entire show in two sentences:
Anne: I'd get this truck packed before the new stuff gets here. Wanna give me a hand?
Angel and Connor's last words from the finale:
Connor: They'll destroy you.
Angel: As long as you're okay, they can't.
In the series finale, Wesley is dying and the demon goddess Illyria pretends to be Fred to comfort him. Illyria had previously offered to do this before the final battle, but Wesley refused because "it would be a lie." As he's dying, Illyria asks if he'd like her to lie to him, and he says yes.
Likewise, near the end of the episode, Illyria is grieving for Wesley, and doesn't understand why she can't stop doing so. Revealing that the god of demons is considerably more human than she thinks she is.
Fridge Logic can kill the moment, though, if you watch the scene again and realize that Illyria could've saved Wesley if she'd arrived five seconds earlier. Which would've been very easy to accomplish if she weren't walking so slowly.
This troper particularly liked the part where the demon who killed Wesley offers Illyria one free shot (thinking her a helpless human girl)—and she pulverizes his skull with one punch. Not exactly Heartwarming, but it made me feel better.
This troper likes to consider that a combination of Heartwarming and Awesome - Illyria, the primodial demon who considers humanity beneath her, kills Vail in retribution for what he did to Wesley.
In the Angel: After the Fall/Angel comics
When the audience finds out what Angel named the dragon. It's Cordelia. Though it doesn't seem so much like it when it's explained.
One scene has Illyria getting angry about those she cares about dying as Ghost!Wes disappears; a few panels later, a sign is shown with "Doyle" written on it.
Connor telling Human!Angel to go home, because he cares whether or not he gets hurt.
Spike:"What was that? If he (ghost!Wes) wasn't see-through, you would have kissed wouldn't you? I got that vibe."
When Angel is reunited with Connor, Angel's so relieved and happy that he's crying. Also when he talks about how much he loves Connor.
One scene has Betta George, a telepathic demon, reveal that Angel and Spike both (simultaneously) thanked each other silently and in another scene that Spike thinks of Angel like an older brother (though he immediately denies this). It's especially heartwarming when you realize before this Angel previously lost the rest of his close friends.
Angel named the research wing of the library after Fred and Wes.
In Only Human when Fred's parents tell Gunn that they will love her no matter that (they still don't know about Illyria).
Illyria staying by Gunn's bedside while he heals and killing his enemies and Angel forgiving him.
"I think you're kind of amazing. And I'm proud to be your father." **Hug**
The end, set against a background of the faces of Wes, Fred, Cordelia, Doyle, and Lorne:
Angel: "I've made mistakes along the way, and I've lost people I will never stop missing. But I will spend the rest of my life, however long that is, making their sacrifices worthwhile. Tonight, I'd like to think they'd be proud of me. I know that I will always be grateful for them."