Rocky's grief over Adrian's death, especially the scene where he confesses to Paulie that he has a "beast inside" him because of it.
Let's face it; even if you know ahead of time Adrian doesn't appear in Rocky Balboa, the sight of Rocky visiting Adrian's grave is absolutely heartbreaking.
And he visits it so much that he keeps a chair in a nearby tree.
And in the final scene of the film the chair is gone.
Rocky confiding to Paulie about a "beast in the basement." To see Paulie go into Sincerity Mode with Rocky as he talks is both heart warming, and tear jerking to see an older Balboa crying.
Paulie also mentioned to Rocky that his meat factory job is all he has left. This makes the lay off much more heart breaking for Paulie, especially after Rocky looks at his eye.
Rocky: ...yo what happen to your eye?
Paulie: I hit it. I deserve it.
In a deleted scene, appropriately titled Paulie breaks down, he breaks into tears about how he was laid off from the meat factory, how poorly he had treated Rocky, and how deeply he misses his sister, and wishes that he had died instead of her. Seeing Paulie broken down like this, especially after how he's been through out the whole series, is sure to break any fan's heart.
Paulie: (to Rocky) "If I'm going to be lonely, I'd rather be lonely with you."
Rocky's comeback against Mason Dixon slowly becomes a tear jerker of the joyful kind, particularly his struggles in the last round (and the Ironic Echo of his speech to his son).
The video montage of all the people running up the steps at the end credits. See it here.
Much of Paulie's "Change the channel from yesterday" speech. On the outside it sounds like he's being his usual Jerkass self, but when you listen to what he says, it's about his regrets for his treatment of Adrian while she was alive.
Paulie: You treated her good! You had the good times! I treated her bad! I don't have to think about this...
Rocky: She always loved you, Paulie.
Paulie: Sorry, Rocco. I can't do this no more.
After the video montage in the end credits of fans running up the steps, the scene shifts to Stallone as Rocky, standing alone on the steps as the somber version of "Going the Distance" from the first film's end credits plays. It drives the point home that this movie is indeed the last one in the main series; even in the Creed movies, Rocky's only a supporting character.
Seeing Rocky run up the very steps he made iconic one last time can be seen as this.
