When the Sesame Street gang sings "Deck the Halls" and it's Oscar's turn, he declares, "I will not sing this song!"...and the others pick up the next line without missing a beat.
The Count singing "I Saw Three Ships".
Most of the interactions between the various cast members from the three Muppet shows.
Animal commenting that Cookie Monster is his kind of guy after chomping down a plate of Christmas cookies.
Janice: Who was that strange, blue creature?
Animal: That my kinda fella. Hahaha.
Ernie & Bert telling Doc that the educational tidbits from Sesame Street (i.e. "Did you know Doc starts with the letter D?") is small talk from where they come from.
Rowlf & Sprocket communicating in barks, made better by how Sprocket is actually barking while Rowlf is just saying "Bark" & "Woof".
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?"
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!