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.
Continuity mistake: When Argyle is sitting in the back of the limousine he hears on the news that terrorists have taking over the Nakatomi building and that their means of communication is through the use of CB radios that they brought with them. Argyle then moves to the front seat of the limousine and turns on the CB which is immediately tuned to that exact channel.
Continuity mistake: Argyle is seen driving the limo around the garage after hearing that the terrorists have taken over the building. During that scene there are a couple of shots where the back window does not show anyone or anything in the back. But after the limo stops, he looks in the rearview mirror and the bear is sitting with it's head over the back of the seat. This would have been noticeable in the driving shots.
Continuity mistake: In the outdoor shot where the lobby is being blown up after John drops the explosives down the elevator shaft, the stairs in front of the building disappear for a brief second.
Visible crew/equipment: When the car and truck drive upto the building, the truck goes down a ramp - another moving shadow can be seen cast over the car, from the camera crew.
Continuity mistake: In the scene where Sgt Powell is entering his patrol car after checking the building, you can see that the spotlight is missing from the car exposing the mounting holes. However, after the body has landed on his car and he retreats rather quickly, the spotlight is back on the car.
Other mistake: McClane's undershirt is white for the beginning of the movie, but when he emerges from the air conditioning ducts, it is now a dark colored shirt (it looks very dark green) with a lot of stains on it. If this is supposed to represent the same shirt, just very dirty, it could not have come from the A/C duct work, for it was all brand new and the interior shots showed it all to be spotlessly clean. The change in appearance is far too abrupt to be possible.
Revealing mistake: When John jumps off the roof with the hose tied around him, you can see that the stuntman's hair is slicked back with fireproof gel during the explosion.