Danny and Sam's first kiss in the Grand Finale. When Butch Hartman finally has a long awaited pairing happen, he doesn't slouch.
Just about everything in the episode "My Brother's Keeper" is heartwarming, but the Crowning Moment of Heartwarming is Jazz giving a sisterly kiss to her brother to cheer him up. After she leaves, Danny gives off a brief smile before he ruins it with an "oh, gross" comment. Still, it was sentimental.
Danny: (to Spectra) Man, I am so tired of you dumping on me. And I am so tired of dumping on myself. Jazz never did that, even when I was mad at her. And I won'tlet her down!
Jazz learning about her brother's secret. Rather than rat him out to their parents or directly confront him about him being half-ghost, she chooses to keep his secret and pretends not to know at all.
The ending was probably the best. After watching Danny be miserable for nearly the entire episode, watching him fly around with a smile is welcoming.
Danny learns Esperanto just to be able to understand Wulf.
The final episode had a few too. Specifically, right before Danny goes off to attempt to save the world and Sam gives him the "Wes" ring back (which hadn't been seen since season 2). The conversation is extremly heartwarming.
Danny: Sam... If we make it through this-
Sam: When we make it through this.
Danny: Right. When we make it through this, I have a few things I need to talk to you about.
Sam: I think I'd be willing to listen. And no matter how this thing ends...this whole ride we've been on together, I wouldn't change it for the world. (Places her hands in his.) Not. One. Bit.
Danny: Me neither. I- (Sam kisses him on the cheek. Danny grabs her chin, tilts it upwards and kisses her on the lips. Cue a beautiful slow rendition of the main theme, Foot Popping, and rotating camera.) ...Wow. Remind me to save the world more often.
Don't forget their second moment at the end of the episode when they become an Official Couple:
Danny: Sam, I could never have done any of this without you. (He slips the ring on her finger. She tears up.) And I don't care what's coming next. I just hope that, whatever it is, you're there to share it with me.
Sam: I will be. I just have to warn you, I'm no pushover, you know. I still have my own way of doing things.
Danny: That's what I'm counting on.
The Fright Before Christmas. When all the villains team up with Danny to take out The Ghost Writer, to make it clear that no one breaks the Christmas Truce and gets away with it. And especially when the villains all help repair the damage to the town so that everyone has a merry Christmas.
Tomorrow you'll see what this baby can do! /... But all ghosts share the truce... even half-ghosts, like you.
Really, any time that Danny and the villains of the show have a moment where they're on good terms. Like in "Girls Night Out", when Johnny 13 escapes the Ghost Zone to blow off steam after having a spat with Kitty, and leaving after Danny tells him to.
The Ultimate Enemy. Just when it looks like Danny is about to fail to save his family and friends from an explosion, Clockwork freezes time and rescues them. Then, he takes Danny back in time so that he can undo his mistake.
Clockwork: You've given everyone else a second chance. Why not you?
In the same episode, when Danny admits to having peeked at the answers to the test. Lancer's smile when he shows Danny the door and tells him that he will have plenty of time to study for the make up test is wonderful. You can tell that he's truly proud, even with the behaviour he usually has. And even earlier in "Teacher of the Year" when he finally gets Danny to take an interest in his studies - even if he had to use a gimmick, it is moments like these that show that Lancer truly does care about his students and wants them to succeed, making him less the monster the show portrays him as for the sake of a gag most of the time.
Heck, even Clockwork has a proud smile at Danny's actions in the end. If that's not heartwarming, I don't know what is.
Danny and Jazz's hug. After everything that's happened, their bond still stands strong and it warms my heart every time I see them hug in this episode.
When Danny's identity is revealed to the entire town and the Guys in White go after him, all of Danny's classmates go out of their way to get him to safety.
Danny: You're helping me?
Dash: Well, duh. You've been saving us for, like, ever.
From "Masters of All Time" we get this touching reunion between Danny's (not) parents:
Jack: I never stopped wanting to see you. I just couldn't face you as the thing you despise: a ghost. But I had to come back because of this. (hands over Danny's wallet photo)
Maddie: A family... with you? (tears up and hugs him) Oh, Jack!
In the season 2 episode Reality Trip's ending, an exposed Danny faces his stern-looking parents, expecting a full berating for lying to them for months. What he gets is them accepting him, smiling and understanding his reasons to lie. Touched as he was, he still wiped their (and the rest of the world's) memories of his reveal.
Maddie: We love you, Danny; you.
Jack: Whether you're ghost, boy, or something in-between!
Freakshow and Lydia's interactions in Reality Trip. In their first appearance, we don't see what becomes of Lydia when Danny destroyed Freakshow's Mind-Control Device, and here we find out why—because Lydia wasn't being mind controlled. Freakshow is shocked and angered when he sees that the Guys In White have her in their custody, frees her as soon as he can and when Freakshow himself is sucked into the Fenton Thermos, Lydia immediately attacks Danny over it (until he scares her off by turning her into different objects). It's strongly implied that Freakshow and Lydia are romantically involved.
In D-Stabilized, when it seems that Danny's attempts to save Dani seemed to have failed... she reforms and comes out of the puddle of ectoplasm in one piece. The two Phantoms embrace.
Dani: I'm me again! (hugs Danny) I knew you wouldn't let me down!
At the end of the episode, when Danielle thanks Danny for saving her, he replies "That's what family's for, cuz!" Right before she lives, she kisses him on the cheek and promises to see him again. It goes to show that despite where she came from and how they met, Danny considers her family and will be there for her. And Danielle, in turn, shows gratitude for everything he's done for her and promises a reunion.
The Dairy King's only appearance in Bitter Reunions, when he frees Danny and proves that not all ghosts are in fact evil.
Dairy King: Well, not all ghosts are evil, right? Some of us just wanna be left alone now, don't ya know?
While it's mostly played for laughs (and quite effectively), the scene near the end of the Reign Storm movie. When (in no particular order) Maddie, Jazz, Mr. Foley, Mrs. Manson, the mayor, civilians, and the police all fight over who's going to fight Pariah Dark, possibly to the death in the Ecto Battlesuit.
Some fridge heartwarming for Bad Future Vlad from Ultimate Enemy. When last we see him he's looking all menacing and commenting that if he kills Danny he could prevent him from ever turning into Dark Danny. When next we see Danny he's flying in just fine, meaning Future Vlad had trusted in Danny to be willing and able to do the right thing and stop his evil future self.
Despite how it blew up in his face, the fact that he took in Danny and cared about him enough to attempt to remove the ghost part of him is touching. And he doesn't insult Jack when he talks about how Danny came to live with him.
Small potatoes compared to most of the examples on this page, but in "Girls Night Out", Skulker going after Danny is motivated entirely by his desire to win Ember's heart back. Aww. On that note, one of Ember's commentsnote "And you told Skulker that he's a lousy hunter!" "Well he is, he can't even find the remote!" implies that they're living together, which is weirdly adorable.