Continuity mistake: When Buddy first meets Jovie, she is wearing much more make-up by the end of the scene. The amount of lipstick increases quite a bit, for example.
Revealing mistake: Towards the beginning, in one of the scenes where Buddy is sitting on Papa-Elf's lap, you can see Bob Newhart is not quite lined up with Buddy, or his legs in the foreground (Which are actually the legs of a little kid), as he is really a short distance away, to make it look as though he is shorter.
Continuity mistake: When Buddy is lying in the roll-out bed in the living room of his father's house and asks to be tucked in, his father does so. His father comes back up and ends up having a red thread on his shirt. The shot switches to Buddy, then back to him and the thread is now on his right arm.
Continuity mistake: When Buddy's dad opens the present from Buddy, he holds it directly above the box. In the next shot, a closer shot, he is holding it next to the box. The following shot shows him holding the present directly above the box again.
Continuity mistake: Watch the shoppers behind Buddy after the Gimbel's supervisor announces Santa's upcoming visit. Without enough time to have moved between shots, a guy with a yellow elf hat is standing where a lady with a black coat has been.
Revealing mistake: The number of snowballs the bullies throw at Michael and Buddy is very unrealistic. There are about five kids, and yet at any given moment, there are dozens, if not a hundred snowballs in the air.
Visible crew/equipment: Before Buddy goes to New York, he is talking to the snowman, and while he is walking with him, you can see multiple shadows. One to his left, right, front, and back at least.