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.
Other mistake: When John and Zeus are being chased by the German guys in the car, John asks Zeus to pull out all the fuses in the fuse-box. The next shot is a quick shot of the speedometer etc. and it is lit up. Surely if all the fuses were out it wouldn't be lit?
Factual error: The security guard at the vault door fires a few shots from his handgun (about 6) down the hall towards Simon and the other terrorists. He then loads one of the shotguns and calls for help. After finding that the front guard was a terrorist he fires 14-15 rounds from the same shotgun before he is killed. The problem is that it was a Remington 870. Like most shotguns that size they only hold 8 rounds, or 9 if one is chambered.
Factual error: The entire plot is based on the idea that they're going to steal $140 billion worth of gold. At a gold price of $300/ounce, that's 14,500 tons of gold - how's the suspension on those dump trucks? There's no way trucks like those could carry over 1,000 tonnes each, but Simon tells McClain there is $13 billion worth of gold in the back of McClain's truck.
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.
Revealing mistake: When Charlie (the bomb technician) hears that there are still children in the school, he continues trying to disarm the bomb. Pay attention to the shots of the countdown timer. It goes straight from 1:01 to 0:59. There is no 1:00.
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.
Factual error: I'm afraid John and Zeus are in for a very nasty surprise when John uses a few drops of the binary explosive to blow apart Zeus's handcuffs. We know the explosive is sensitive to shock or impact, like nitroglycerine - we see samples being detonated by being hit. So, when John detonates those few drops it will set up an adiabatic shock wave, and since the binaries have mixed at this stage that is going to set the whole lot off. It's called a sympathetic explosion. Those two tiny drops are certainly enough - they couldn't possibly cut hardened steel if they weren't.