This scene becomes funnier if you were a naive young child who attempted to cut the same snowflake pattern that he has. If you pause the film, get some paper and scissors, and copy exactly what he does, what you end up with looks absolutely nothing like that snowflake.
And shortly before, Jack tries electrolysis on a candy cane and ends up with a limp candy cane with no stripes, and his reaction is much the same as with the snowflake.
The part where Sally is worriedly talking to herself while Jack is practicing his "ho ho ho" in the background.
"A pox? How delightful, a pox!"
"If you please..."
The street band attempting to play "Jingle Bells", with tritones for the harmony, and later their rendition of "Here Comes Santa Claus" in the same style. Evidently, those guys do not know any major keys.
When Lock, Shock and Barrel try to stuff Santa Claus down the chute leading to Oogie's lair, he gets stuck.
Shock: "I think he might be too big!" Lock: "No he's not! If he can go down a chimney, he can fit. Down. HERE!"
And when we see an outside view of the chute as Santa grunts and cries out as he goes down it, knocking rivets out on the way.
And also right after they first shove him in the opening of the pipe and realize he is stuck we get a shot of Santa's mistletoe underwear. How humiliating for Santa.
As Santa is shoved into the opening of the pipe he makes a funny face by rolling his eyes in two different directions.
Also Lock, Shock and Barrel smacking Santa's butt with the plunger, pitchfork and broom.
The way he says "Would someone SHUT this fella up?"
The start of the deleted verse ("Well, if I'm feeling antsy...") The pun is funny enough, but there's something really hilarious about his casual, cheerful tone when he says this line. It just hammers in how Faux Affably Evil he is.