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: 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.
Continuity mistake: In the taxi scene were you see the ambulance escort pull onto the road; you see John swing around a corner going very fast, when Zeus says "slow the fuck down McClane". He gets ready to pass a large black car but the next shot (an in car shot) you can see that he never passes the car.
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: 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 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.