Continuity mistake: After he climbs out of the air duct, John McClane's shirt goes from white to green. The dirt and grime from the air duct would cause discolouration only where the shirt comes in contact with the duct. However, his shirt is uniformly green including places that wouldn't make contact and it is impossible for all surfaces to make contact even in small areas. (00:23:05 - 00:56:30)
Continuity mistake: When Frank sees what he thinks is the Ghost of Christmas Future in the elevator, he flips out and accidentally clocks Grace in the jaw. She puts her right hand to her face, but in the next shot she's holding her left hand up.
Continuity mistake: The strings of lights for the reindeer reins change continuously from one strand to two.