The ending to "Dog Tags" is surprisingly sweet for this show; when Ren can't gain access to his own dog club, Stimpy gets the idea to bring him to a club for cats. Ren (dressed as a cat) and Stimpy sing "When Irish Eyes Are Smiling" with some other cats and then share a bowl of milk together. Awwwww.
Fake Dad. Ren CANNOT bring himself to spank Kowalski at the very end, and is reduced to tears when he's taken back to prison.
Pilot episode "Big House Blues". Ren and Stimpy get put into shelter by a dog catcher and a little girl chooses Ren for her pet. After he's taken out of the cage, he mumbles at the girl "If you take me, you have to take Stimpy too!".
"Stimpy's Fan Club". Ren is overjoyed to finally get a fan letter for him, after a stream of fan mail for Stimpy. He gloats about it and shows it off... only to find out the letter was from Stimpy himself. Extra points as he glances dumbfounded at Stimpy, who despite being verbally and physically chewed out by Ren only seconds prior, just gazes back with a kind, forgiving smile and hugs him.
Stimpy: I meant every word.
Ren (bursting out crying and snuggling up to Stimpy): Look at me! Big, mean Ren, crying like a baby! Pretty pathethic, huh?!
Stimpy: Oh, peeshaw! No one will ever know. You just cry your little eyes out.
Even better there's a meta example for this. The next week, Nickelodeon got a truckload of letters from children telling Ren they loved him too. Aww...
"Son of Stimpy". Crowning Moment of Heartwarming combined with a Crowning Moment of Funny, since the whole episode is dedicated to Stimpy's fart, that he happens to believe is his son (originally aptly titled "Stimpy's First Fart"). Stinky (as in, the fart) leaves his home (as in, Stimpy's butt). The whole episode is spent on Stimpy trying to find Stinky, and is sealed with a heartwarming reunion.
Don't forget Ren. His grief at losing Stimpy and his joy at having him back are downright beautiful, especially considering it's Ren we're talking about.
The scene of a completely deflated Stimpy, who Ren vigorously tries to cheer up with his favorite programs (all of which he has shown nothing but contempt for beforehand). He that desperately wants Stimpy to be happy again.
The end of Stimpy's bedtime story "The Littlest Giant". The eponymous giant saves a village from a terrible drought. In return, he gets what he always wanted: a friend.
"Mad Dog Hoek": During Stimpy's "I like Darren!" rant, he goes nuts. It was rather heartwarming seeing Ren who's normally rather crazy and rude try to calm Stimpy down.
I forgot the name, but in the episode where Ren and Stimpy are marooned and Ren first runs into the whale's corpse, his reaction is to jump into Stimpy's arms and hold him while squeaking in an adorably pathetic and panicked manner. Yes, Ren can be cute. 
It's "Aloha Hoek" I believe
The beginning to Man's Best Friend (Yes, the one that got John K. fired), where George Liquor sees Ren and Stimpy in the pet shop, asleep on top of each other. And it's incredibly cute. Liquor ruins the moment, though. They do it again in the fish bowl.
It could be said that Man's Best Friend's ending was just as heartwarming. Just after Ren beats the heck out of George Liquor, George grabs him by the head in unbelievable rage. Then all of a sudden, they chew rubber craps while dancing to Raymond Scott Music.