The Ice Skating scene has plenty of funny moments, crossing with heart warming ones too.
Adrian: Why do you fight?
Rocky: Because I can't sing or dance.
Adrian is clearly embarrassed at being alone with Rocky in his apartment.
Rocky: I can call your brother if that's what you mean. (Opens up the window and yells) Yo Paulie, your sister is here with me! I'll call you later, all right?
A Brick Joke. Before the match, Adrian was wearing a hat. When Rocky cries out to her and Adrian finds him in the ring...
Rocky: ...yo, where's your hat?
A Call Back to the first movie where Gazzo's handler insultingly suggests that Rocky should take Adrian to the zoo because "it's for retards!" Seeming unintentional, but Rocky ended up proposing to Adrian at the zoo.
This exchange between Gazzo and Rocky.
Gazzo: "Interested in a Condominium?"
Rocky: "Condominiums? Never used 'em!'"
Becomes extra funny when Rocky and Adrian discovers that they're going to have a baby.
"THE STALLION CHICKEN" Funnier when...
Mickey trains Rocky making him catch a rooster to develop his speed (Mickey: Ya catch this thing, ya can catch greased lightning!) After a while a frustrated Rock:
"I feel like a Kentucky Fried idiot!"
Before he chases it we get this exchange:
Mickey: Do ya have to wear that stinkin' sweatshirt?
Rocky: It brings me luck ya know?
Mickey: You know what it brings? It brings flies!
When deciding how Rocky and Adrian should name their kid...
Rocky: What should we name him?
Paulie:Paulie's a good name.
Rocky: ...yeah, Paulie is a good name.
After Paulie's ranting to Rocky, and Rocky giving him a Reason You Suck Speech, Paulie tries to assault Rocky, only to give up and ask
Paulie: ...can I have a job?
Rocky: ...all you had to do was ask.
At the beginning of the charity match, Rocky asks Thunderlips for an after match Polaroid. Lips simply shoved and snarled at him. When the match finally concludes in a draw, Thunderlips is hilariously much more calmer.
Rocky: Yo, why you so rough out there?
Thunderlips: (Much more calmed down) That's the name of the game.
Rocky: ...well while we're at it, mind that Polaroid I asked for?
Rocky: You know, sometimes, charity hurts.
Mickey gets a load of memorable lines during Rocky's exhibition with Thunderlips during Rocky III, but perhaps the best is one where Rocky, stunned at the size of Thunderlips, asks what a guy eats to get that big. Mickey immediately quips "About 202 pounds." Right afterwards, the announcer mentions that Rocky weighed in at 202 pounds for the bout. Something about the way Mickey starts chuckling at his own wit just seals it.
Rocky throwing Paulie in the pool during training.
Half way in the movie, Paulie and Apollo were at odds, particularly about how Paulie doesn't respect Apollo's home town. Near the end, they develop a mutual respect, to the point where Apollo kisses Paulie on the cheek during the match.
When Apollo tells Rocky to keep it cool when Lang started insulting him, Lang insults Apollo, where he then turns his back, prompting Lang to tackle him, nearly starting a fight between them.
Firstly, Paulie's initial reaction, clearly disturbed by the robot. Then he figured out how to change it's voice and mannerisms to reflect that of an ideal wife of his. He lampshades that too.
"When I come back, that thing will be like my wife."
Apollo's reaction to seeing that Robot. His face seals it.
"...... umm, anyway, as I was saying"
After that heart warming speech Paulie gave to Rocky, and listening to the Russian National Anthem, Paulie follows up the speech with this exchange.
Paulie: Remember when I said I wanted to be like you?
Paulie: Forget it.
Rocky: ...Thanks, Paulie.
Something of a meta example: many fans were making jokes prior to Balboa coming out about how Rocky's opponent was in a no-win situation - either he'd beat up a sixty-year-old man or he'd be beaten up by a sixty-year-old man. In the film itself, a reporter says almost those exact words.
In the deleted scene: Breakfast has Rocky attempting to consume raw eggs, only to spit em out afterward.
Rocky first sees a black and white kid hanging out at the street corner, and Marie mentions that one of them is her son.
Rocky: You know he sort of resembles you, he got that, big irish hair.
Marie: Yeah, it's the other one.
Rocky and Spider years after their first and only fight. More or less heartwarming, because he offers Spider free meals at his restaurant and even more heartwarming because Spider wants to work to earn those meals. But the exchange between them is hilarious ending with this line.