Richard Harrow winning over Margaret's kids, who were scared of him due to his appearance, by saying he's the Tin Woodsman, and later Margaret apologizing for her initial reaction to him.
Richard: The Tin Woodsman. That's me. (He taps the tin mask that covers his battle scars.) Maybe I need some oil.
(The children giggle.)
Emily: Mama, is it really him?
Margaret: Why, yes dear, directly from Oz. And who better to have in our house than the mighty Tin Woodsman?
The look on Nucky's face when Margaret tells him that the children "miss their Uncle Nucky".
Al Capone, of all people, emotionally talking about trying to help his deaf son feel the vibrations of music but knowing he doesn't understand. Also may count as a Tear Jerker for some.
Al's moment of maturity in "The Emerald City", taking the rabbi's advice about accepting responsibility for one's actions to heart and apologizing to Torrio.
When Margaret is debating whether to become Nucky's kept woman in exchange for support for her children, she goes to the head of the Woman's Temperance League for advice. One might have expected a woman in such a position to be an uptight, Bible-thumping prude, but she is instead completely understanding, tells Margaret to do whatever she sees fit, and tells Margaret she isn't answerable to anyone else for her decision. The only advice she gives her is that she should take a pamphlet containing instructions on how to avoid unwanted pregnancies.
Eddie "Izzie" Cantor coming to cheer Lucy up while she's pregnant, confined and completely miserable. While many of his attempts to make her laugh go over her head, he perseveres until he finds a form of joke simple enough to make her smile. This is particularly notable since Lucy has been completely abandoned by all her "friends" from her past life, and this act by the famous and high-society Izzie is, purely and simply, him doing something nice for somebody else without any financial, political, social or sexual motivation.
Van Alden getting a record player for Lucy.
Richard allowing Angela to paint him without his mask, and Angela's complete lack of horror when she sees his face.
Jimmy and Richard in "Gimcrack and Bunkum", just after Richard has attempted suicide:
Richard: Would you fight for me?
Jimmy: Down to the last bullet.
Richard: Then let's go to work.
In the same episode, Richard finding a pair of friendly hunters and their old dog during a trip to the woods who convince him to not kill himself.
Lucy kissing her newborn daughter on the forehead in "The Age of Reason".
Lucy: I did it all myself.
Teddy finally calling Nucky "Dad".
Margaret breaking the hospital quarantine to curl up in bed with her daughter in "Battle of the Century".
The Commodore coming to Gillian's defence when a drugged and enraged Jimmy tries to strangle her in Under God's Power She Flourishes (in addition to being a Moment Of Awesome,) shows that, while he may have crossed the Moral Event Horizon when he raped her when she was 13, he truly does care for her. Especially powerful considering he was still recovering from a stroke, and there was an earlier episode where she had taken advantage of his weakened state to slap him repeatedly for what he did.
Jimmy and Richard drinking and telling war stories in "To the Lost". It's heartbreaking when you know what's about to happen, but seeing the two soldiers laughing and relaxing for once is sweet.
Jimmy taking his son for a pony ride, and Tommy's response to the question of whether he wants a soldier or cowboy hat:
Another scene in "To the Lost", when Margaret sees Nucky and Teddy encouraging Emily to walk with her leg braces on. It's the moment that makes Margaret decide to marry Nucky.
Richard taking care of Tommy after his parents' death and trying desperately to keep their memory, especially Angela alive in him despite Gillian's efforts.
Seeing Manny Horvitz's unexpectedly sweet and loving relationship with his wife.
Eli's homecoming, with a huge group hug from his kids, followed by one with his wife.
After a misguided attempt to toughen up his son after learning he's being bullied, Capone tears up along with him and they just hug. And later on, he goes after the thug who beat up his friend the instant he hears about it.
And later that night, he returns home to his son and plays "My Buddy" for him on the mandolin, knuckles still torn up from killing the thug in the bar. It's the whispered "good night" after the cut to black that made this troper bawl.
Let's just be clear here. Al Capone savagely beat a man to death. And it was a Crowning Moment of Heartwarming.
In "You'd Be Surprised", Charlie's been summoned by Joe Masseria. Meyer, even knowing Masseria hates Jews, wants to go with him, and warns him not to sit by the window (essentially telling him not to make himself a target).
It's somewhat heartwarming that while Meyer Lansky is very much a Combat Pragmatist, and more than once talks his way out of a shootout or his own execution, but when Masseria's thugs accost Benny, Meyer immediately runs outside, gun out.
Nucky and Willie helping Emily in the Easter egg hunt.
Nucky and Eli reflecting on old times and expressing concern for each other's families while fixing a car. One season after the two of them had gone to war with each other, it's nice to see that they trust each other again.
Richard's entire love interest subplot with Julia, from the beginning when they first meet and she looks unblinkingly at his face as they shook hands (much like Jimmy did when they first met) to the Easter dinner where she prepares a separate dish for him so he could eat in private to the "family" picture with Tommy at the fair to the their dance and kiss at the American Legion to their night together on the beach and under the boardwalk.
Tommy's interaction with the friendly prostitute is surprisingly sweet - until the inevitable happens.
Nucky might treat Eddie like crap, but when Eddie is shot in the penultimate episode of Season 3 because he refuses to leave Nucky's side, Nucky risks his life to get Eddie to safety and given treatment.
And Eddie returns the favor: Throughout the episode he deliriously keeps talking in German to Nucky, and when he's finally lucid enough to translate it, it turns out he was saying a Rudyard Kipling quote about how you can succeed if you don't allow yourself to panic. Even in intense pain and delusion, he's trying to help Nucky.
Nucky being clearly ashamed of not knowing if Eddie has family.
Chalky sticking with Nucky and refusing to sell him out, despite Nucky's rather dismissive treatment of him throughout the season. Related to this and the above examples, due to his ordeal and Eddie and Chalky's loyalty, Nucky seems to finally "get" the problems with his usual behavior (treating people badly and taking them for granted in between throwing money at them), and shows visible gratitude to Eddie and Chalky and regret for his own actions.
In the final scene of "Two Imposters", when what appears to be Rosetti's crew (it's actually Eli with reinforcements from Chicago) pulls into the lumber yard where Eli's eldest son is sheltering Nucky and Eddie, Nucky's immediate reaction is to walk directly in front of Eli's son.
After spending the whole season looking like nothing but a Jerkass, Paul Sagorsky recognizes a fellow soldier who's just come from a war when Richard shows up with Tommy, and implicitly gives his blessing to Richard and his daughter by simply telling him to clean himself up before he comes home. He even puts Tommy up in his dead son's room.
After Richard comes out of a Put Down Your Gun and Step Away situation with Tommy, the two of them share a hug, showing just how much Tommy has viewed Richard as a father figure.
In the season premiere, Richard Harrow cuts a swathe of violence across the country, and finally reaches his destination: his old house, where his sister still lives.
Nucky making a point of thanking Eddie, an indication that his realization of how much he took him for granted stuck.
Richard being unable to shoot his dying boyhood dog, Sampson.
Emma embracing Richard and telling him "you must call yourself to account".
Tommy goes on a long tirade about astronomy upon meeting Richard again (in contrast to his mute encounter with Gillian in the previous episode), and when he asks if Richard is coming home with them, Julia (who has been understandably bothered by Richard turning her life upside down and then disappearing for months) says that "He's coming."
Julia and Richard have the cutest and most awkward proposing scene ever.
After Al survives a shooting by anonymous hitmen, his first immediate thought is towards his older brother as he calls his name out in pained desperation.