Continuity mistake: When John and Zeus use the cable to get down onto the boat the cable hits one of the guards and kills him - as it does that he fires his gun, but when Zeus picks up the gun and tries to use it later, the safety is on.
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?
Stupidity: In the scene with the water-jug puzzle, both Zeus Carver and John McClane initially suspect that the small carrying case contains a bomb. McClane goes ahead and opens the case, which confirms that it is a bomb (the electronic readout in the case even says: "I AM A BOMB. YOU HAVE JUST ARMED ME"). If they suspected it was a bomb in the first place, why didn't Carver and McClane immediately try to warn the dozens of pedestrians all around them to evacuate the area before McClane opened it? That would be a seasoned police officer's first instinct. Instead, McClane and Carver banter and bicker and never attempt to warn the public, even as the timer is ticking down.