The sequence when Jack delivers all the presents. All those demented, evil toys, and that snake thing that ate the tree...
The snake thing can also be seen trying to eat one of the monsters during "Making Christmas".
The shrunken head that one kid held up. It didn't help it played during one preview on t.v. a lot. He looked just as frightened as confused as real life kids did.
The Clown with the Tear-Away Face. Is it the fact that he tore off his face, the whole being-a-clown thing, or the way his voice changes? There's something so specifically freaky about that character that you wonder if it wasn't inspired by an actual nightmare of Burton's (or Selick's, or Elfman's...).
Add to this the fact that they are not only terrified of him but serve him anyway, and the fact that they are implied during the ending (and much of the film, really) to be mostly harmless tricksters, and you begun to wonder just what Oogie did or said to get the trio THAT MUCH under his thrall.
The end of Oogie Boogie's song when he goes from bright green to dark gray and then he did that demonic laugh. Yikes!
When Jack gets to the end of the Town Meeting song, and sings "Sandy Claws" he does a rather frightening face that fills up the screen.
"Kidnap the Sandy Claws, throw him in a box, bury him for ninety years, then see if he talks!"
"We're his little henchmen, and we do our job with pride. We do our best to please him and stay on his good side."
Evil 'lalala's of evilness... *shudders*
If you think the movie version is scary, listen to the Korn cover.
That poor bug they trapped. First they dunk it in something that turns it green, then they send it down to Oogie Boogie. It has time for a terrified squeak before it's sucked out of the cage and devoured.
Fridge Brilliance: Glow-in-the-dark paint often looks green under normal light, and Oogie Boogie is made of bugs that glow in the dark...
Seeing Santa tied up and helpless on top of that table-platform thingy as the black lights go up, the bats screech, and then Oogie Boogie walks in, and you know he's up to no good.
The mere thought of anything terrible happening to Santa Claus is terrifying to anyone, but children will probably be especially horrified that the person threatening to torture and then murder Santa is the Boogie Man. Santa Claus, a famous childhood icon, is being tortured by the Boogie Man, a famous childhood fear. Now that's terrifying.
That alternate scene where instead of a pumpkin, the vampires are playing ice hockey with a severed head (Tim Burton's in fact). No wonder they cut it.
And this thing is supposed to be hiding under the viewer's bed. Since he's a Halloween Town resident, he probably doesn't mean any real harm, but it's still really unsettling that a creature would be lurking down there, waiting to come out and scare you out of your wits.
The scene where Jack is sneaking around Christmas Town like some giant, suit-wearing spider is intensely creepy and out-of-place in the cheerful setting, despite the happy-go-lucky soundtrack to the scene. The bit where he crawls through a window UPSIDE DOWN and lands in a normal-looking crouch without visibly changing his position especially stands out.
Some might not find it frightening as Jack is just exploring the world of Christmas Town. True, he would easily stick out like a sore thumb in a place like that, but he doesn't mean any harm.