Continuity mistake: When Papa is showing Buddy the sled at the beginning of the film we see the Clausometer (the middle odometer)and the indicator is pointing to the left at around 15 (9pm position). As Papa gets in the indicator is now pointing at around 35-40 (10pm position). Later in the scene when it show a closeup of the meter it is back to 15 and then the last shot of it its 35-40 on the meter. The changes are instantaneous between shots and there is not enough time for them to move that much.
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: 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.
Audio problem: When Buddy asks Jovie out for food, as Buddy and Michael leave you can hear a male voice say "It's okay" but it there's no male close enough to whisper. (Nor a reason for anyone to).
Visible crew/equipment: In the scene in Central Park when Santa's sleigh has crashed and Buddy is repairing the engine, the line that is holding the reindeer in place is visible on the lead reindeer. Not the harnesses that are around each reindeer but the line that is keeping the lead reindeer leashed to the ground is visible.
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.
Visible crew/equipment: When Michael comes in the house after Buddy leaves, and says "Hey Buddy", the reflection of the cameraman is visible in the snow glass on top of the Etch-A-Sketch.
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.
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.