Deliberate mistake: For Hans Gruber's iconic death scene, Alan Rickman's fall was filmed at high speed (for slow-motion playback) against a green screen, and the skyscraper perspective footage was added later as background. However, while Rickman falls away from the camera in slow motion, papers are fluttering around him in the background at normal speed. This was done deliberately to make the shot even more surreal.
Deliberate mistake: In the end scene at the TV set, Scrooge tosses a coin out the window, and the camera follows the coin as it falls in slow motion. The people in the control room also watch this on their monitors. Problem is, this is a live broadcast and filming/watching something live in slow motion is impossible.