Continuity mistake: When the two heroes are crashing through the city in the taxi, the car takes a significant beating. The bumper is pushed up from rough landings and there are long scrapes down each side. When Zeus skids to a halt outside Wall Street station the taxi is in almost perfect condition.
Continuity mistake: When McClane and Zeus are chased after escaping from the aqueduct, look at the right front light of the Dodge Ram the bad guys are driving. It goes from being broken to intact several times. Also, when McClane and Zeus stop the same Dodge Ram afterwards on the bridge, both front lights have magically become intact again and even though now the hood is missing, it's still obvious that the whole front part of the truck is significantly less damaged than before.
Continuity mistake: In the park with the water riddle, just before they argue about Zeus being a racist, you see both jugs. The 3 gallon jug is empty and the 5 gallon is possibly 3 gallons full. Then they argue. By the end of the argument, John says that the 3 gallon jug is exactly 2 gallons full and you see that the 5 gallon jug is full up, giving them the answer to the riddle.
Continuity mistake: When they first start with the gold operation, Targo jumps on a large orange vehicle with some kind of drill on the front, already raised into the vertical position. He then jumps off and in the very next shot the drill raises from a horizontal to vertical position.
Continuity mistake: Towards the end, when Bruce Willis notices the writing on the aspirin bottle, he leaves it in the phone booth and the camera jumps towards it a couple of times. In the first shot the writing's almost vertical, but in the following shots it's at a definite angle. (01:48:35)
Continuity mistake: While in the police station, a chubby guy brings the suitcase with the liquid bomb and to explain how it works he dips a stick with the liquid and throws it away to make it explode. Watch now the guy behind him covering his hears: Headset off, then he puts it on; then suddenly, in a very brief shot, when he is behind the chubby man, the headset is off again. In the next shots he has it on.
Continuity mistake: In Harlem, when Zeus gets his arm cut by the black dude wielding the switchblade, another black dude is swinging the bottle toward McClane's head-with his left arm. The following scene, McClane gets his head injured from the same bottle wielded by the same dude-only this time wielded by his right arm not his left.
Continuity mistake: During the car chase between the Ram and the Mercedes, the Mercedes is going around a red Plymouth Acclaim, but in the very next shot it is a Chevy Cavalier. The Ram truck then starts scraping between the wall and the other side of the Cavalier, but when it cuts to a close-up it is now hitting the side of a grey Buick that was in front of the Cavalier.
Continuity mistake: When they're on the bridge, trying to find a way onto the barge and they first get out of the truck, the skies are very overcast, as shown by the rain on the bridge. When they cut away and cut back to McClane, the sun is now out and very few clouds in the sky. It's fairly obvious that it was shot on different days and the bridge watered down.