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.
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: When they drive through the park and jump off the rock they land on a pile of leaves which covers the hood. Some of those leaves fall off but not all of them - in the next shot the hood is completely clean.
Continuity mistake: There is a scene where Samuel L. Jackson and Bruce Willis are in a crowded park, and they have to defuse a bomb by measuring out water. Before they do that, he picks up a cell phone and calls a bad guy with it. When he is beginning the conversation, the cell phone antenna is down. They change to a different angle, and the antenna is up.
Continuity mistake: When Willis washes the blood off his face after shooting the bad guy in the elevator, the "towel" he wipes his face has no blood on it at all. Even after washing the worst off with the water it would still be full of blood.
Continuity mistake: When Samuel L Jackson runs down the stairs in the subway station, he checks the clock, which reads 10:19:10. Then when the train destroys the clock a bit later, the time's now 10:20:24, though well over 2 minutes has passed (only counting Samuel L Jackson's scenes, as Bruce Willis' presumably take place simultaneously. Counting them separately it's even longer). (00:33:00 - 00:36:30)
Continuity mistake: When Bruce and Sam are driving through the park and jump the concrete ramp out into the street, they land (or drive over) 2 taxi cabs parked there. In the close-up shot of their taxi crushing the front and rear of the other 2 parked ones, you can see that an entire wheel disconnects from one of the parked taxis and begins to fling out into the street with great speed. In the following wide-angle shot of their taxi swerving into the road, that wheel is nowhere to be seen. It should have bounced across the road.
Continuity mistake: Near the end, after the huge gold heist, Simon Peter Gruber and his psychotic lover, Katya, engage in rough celebratory sex, ripping off each other's clothing, such that Katya's upper torso and black lace brassiere are fully exposed. A few minutes later, when John McClain interrupts their sex, Katya's shirt is fully closed.