Revealing mistake: When the driver is taking Amanda to the airport at the end, all the dials on the car are at zero, so obviously the car is not moving at all.
Continuity mistake: The first movie shows Santa slipping on the roof and kicking up a white snow blanket as he falls. The flashback in the third movie does not show the blanket being kicked up as Santa slips.
Revealing mistake: You can see Buddy and Finch's stunt doubles in the ice skating scene. And when they are showing a close up of them you can tell that they are not really ice skating, but just doing the ice skating movement.
Visible crew/equipment: A boom mic is visible at the top of the screen in one of the alternate endings where a mortician comes to get the body of Billy from the coroner.