Tony showing genuine concern and warmth towards Melfi around the time when Junior was their biggest threat, shown in a scene in which he meets with her in Melfi's car as an impromptu therapy session and to obtain information.
Paulie's relationship with his mother. He dotes on her and ensures she is happy and well taken care of and is infuriated by Cookie and Minn constantly picking on her. When Minn gets into an accident due to her crappy eyesight Paulie forbids her from driving her mother anywhere and drives them to their Broadway show and to a restaurant, where he stops the other ladies from hogging the dinner rolls. Turns deadly when he breaks into Minn's house to rob her (he knew she kept all of her money under her mattress due to conversation at the aforementioned lunch) and ends up murdering her with his bare hands. Even when it's revealed Nucci is his aunt rather than his mother Paulie, though initially devastated, still loves her because she is the only person that really loves him and took care of him when he was a troubled youth.
Johnny Sack's death, while certainly a Tear Jerker, is heartwarming by virtue of the fact that his genuine love for his family paid off, and he got to pass away peacefully in a hospital bed surrounded by his loved ones - a luxury not afforded to most mobsters. Especially considering the very close and oblivious brush with death he had earlier in the series, when his beef with Ralph (also rooted in John's love for his wife) almost led to his brutal demise at the hands of the Atwell Avenue Boys.
John's unconditional love of Ginny and his devotion to her despite her being morbidly obese is very heartwarming in particular. He comes home and sees her sneaking junk food. He's incensed at her. Not because her diets are failing, but because she lied to him. He loves her no matter what her size is.
Ginny: "I know I've gained weight these last few years. I see the other wives, the way men look at them..."
John: "Don't I look at you that way? Haven't I always? It was your idea, all this dieting nonsense."
Ginny: "I wanted you to be proud of me."
John: "I am proud of you. I love you."
Tony, at least temporarily, ends his therapy sessions with Dr. Melfi in "Calling All Cars". He acknowledges that he really hasn't improved or grown fundamentally as a person in the four years they've been meeting, and that further attempts would be futile. Melfi, for her part, while protesting Tony's denials that the therapy has helped, nevertheless respects Tony's decision and wishes him the best. Tony gives Melfi his heartfelt thanks and gives her a chaste, yet very tender kiss on the cheek. This very amicable and sweet parting of the ways between Tony Soprano and Jennifer Melfi would have been a far more congenial and upbeat ending of their professional relationship than the bitter, acrimonious break we would see in "The Blue Comet".
The extremely passionate, yet ultimately non-consumated affair between Carmela and Furio is nonetheless permeated with many instances of tenderness, yearning and emotional depth. The conflict Furio felt when given the opportunity to kill Tony and get off scot free, clearing the way for him and Carm to be together, is clear and palpable on his face. As is Carmela's anguish when she hears that Furio has returned to Italy; she looks literally deathly ill.
The reactions (particularly from Carmela and Meadow) to Tony getting shot by Junior. One could make the case that Meadow's pleas for Tony not to die are what finally help in spurring his revival.
In Season 6, there's the moment where Tony comes to the rescue to save AJ from his attempted suicide by drowning in the swimming pool, and then comforting his crying son afterwards. Later on, Tony informs his captains of this, and all of them do their best to comfort Tony, with most of them revealing that their children struggle with mental health too.
In the final episode, Tony's last bit of business on Earth is not any sort of criminal venture, but an effort to obtain Junior's money so that Bobby Baccala's kids can inherit it.
Tony's testimonial to his childhood friendship with Tony Blundetto in "Rat Pack" is heartwarming on yet another, more subtle level. Tony S. relates how he complained about having two sisters but no brother as a child, to which his mother Livia says "What's wrong with your cousin Tony?", pointing out that the bond "Tony Uncle Johnny" and "Tony Uncle Al" had was just as strong, if not stronger, than any actual brotherly link. Even Livia, who throughout the series has been depicted as hateful, manipulative, and utterly devoid of empathetic feeling, was still perceptive enough to appreciate the depth and strength of the two Tonys' true brotherly love.
Tony in "Remember When," when he realizes that Paulie keeping the painting of Tony as a Napoleonic General wasn't a troll after all, that it was a sign of genuine respect and esteem, and it obviously touches him.
Edie Falco's tribute to James Gandolfini at the 2013 Emmy Awards, the first to be held after his death.
Dr. Melfi breaks down in tears during one of Tonys therapy sessions following her rape. Tony immediately gets up and comforts her, genuinely worried to see her so distressed.