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
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."
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.
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.
Edie Falco's tribute to James Gandolfini at the 2013 Emmy Awards, the first to be held after his death.