Revealing mistake: While we see Buddy drinking the entire bottle of Coke, it is quite obvious that he is not actually drinking it. Buddy has his lips firmly sealed around the end of the bottle and gulps down the entire lot. If he was drinking it, the bottle would have caved in as no air was getting inside. It was not even possible for him to blow air from his nose back into the bottle due to the fact that he is continuously gulping.
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.