Doc: Don't you just hate it when you can't speak the language?
Big Bird being blissfully unaware of the Swedish Chef trying to cook him.
Swedish Chef: Gobbla Gobbla Humunga!
The Running Gag where various characters accidentally slip on an icy patch.
Mrs. Bear, after confirming that everyone has arrived at the farm, announces that they've just run out of room. And two of them will have to sleep on coat hangers. Cut to Gonzo and Animal doing just that.
Jim Henson's cameo is generally remembered by the fandom as a Heartwarming Moment, but it ended with this cutely funny line:
Jim: But, tell you what, somebody'll have to do something about these dishes, Sprocket. C'mon. I'll wash, you dry, okay?
Also watch Sprocket during that line. He's smiling and wagging his tail right up until the moment Jim mentions the dishes.
This exchange at the beginning of the film during the car ride up to Mrs. Bear's farm:
Fozzie: Guys, this is the best idea we've had in years! Christmas at my mother's farm.
Kermit: Sounds great, Fozzie, but don't you think we should have called your mother to let her know we were coming?
Fozzie: Oh, how little you understand bears, Kermit! My mother loves surprises!
Gonzo: Good thing; she's in for a beauty.
Swedish Chef: Dis make a gobbla-gobbla faire furnousse!