The episode "A Very Special Episode", as it is one of the few episodes where Beavis and Butt-Head have a Pet the Dog moment.
Although said moment is subverted as the two wanted the baby bird to die and only fed it because they thought it wouldn't die until after it was fed, leading them to nurse it back to health by complete accident.
A more fitting Pet the Dog moment for the two would be the episode Skin Trade where Beavis forms a bond with a dead animal he found at the dump that he treats like a pet. Twisted? Yes, but completely in character.
In a twisted way, "Most Wanted." They boys come across a knife-toting psychopath on the run from the cops in their backyard. Just by being their blissfully idiotic selves, they manage to make him laugh, feel a little better about himself, and turn himself back in to police. And before he leaves, he even gives them prison tattoos as a parting gift! note On their butts. Of butts, with butt-shaped tattoos on them.
"Beavis And Butt-head Are Dead," the final episode from 1997, is a loving tribute (via Clip Show) to the boys and their legendarily stupid antics over the years. According to Word of God, they did consider killing them off, but they were such a labor of love by the producers that they couldn't bring themselves to do it. The episode instead ends with them walking into the sunset laughing to themselves, cracking about Principal McVicker. Cue closing credits, set to Bing Crosby's "I'll Be Seeing You" (set over such touching clips as Beavis scratching his itchy wiener with a Burger World spatula).
In-universe, there's Van Driesen reminiscing about his experiences with Beavis and Butthead while theyre still belived to be dead, showing that not only does he hold no ire for the various physical injuries and pain theyve caused him, he still holds them in the same regard he holds for all of his students.
Van Driesen: "Beavis and Butthead did have a difficult journey through this life. I can only hope that now theyve found whatever it was they were looking for"
In "The Great Cornholio", Beavis actually apologizes to Principal McVicker for the trouble he's caused as Cornholio. McVicker was shocked so much, that he decides not to punish him and actually tells Beavis that he's proud of him.
From the revival season, the episode "The Rat." Despite spending the first half of the episode trying to kill the eponymous rodent that's invaded their kitchen, the dim-witted duo—Beavis, specifically—end up saving its life by accident...and then, after the rat grows visibly attached to Beavis, he suggests that they keep it as a pet. Butt-Head's response? A rather nonchalant, "Uh, okay."
And despite breaking health regulations by taking it to Burger World, Beavis and Butthead do seem to like having the rat around. It's even still alive by the end of the episode!