Corrected entry: When Bruce Willis throws the C4 (stuck with a chair and a computer screen) down the elevator shaft, the C4 explodes. C4 is very manageable and won't explode through physical force; even a bullet won't make it detonate. You'd need the blaster caps or some other explosion to set the C4 off. For more info on C4, read this: http://science.howstuffworks.com/c-42.htm.
Corrected entry: It's been pointed out that McClane's shirt changes from white to green after he emerges from the vents; this is certainly a "cheat" by the wardrobe department to make him appear grimy and battle-worn. But notice, also, that toward the end of the film, the shirt switches back to white.
Corrected entry: It seems to me that alerting the enemy to his presence with the "ho ho ho" stunt was a pretty stupid thing to do and not what a trained officer would do. Without this, the terrorists may have believed that the new fire alarm system had malfunctioned but this made it clear that he was there.
Corrected entry: Towards the beginning of the film, John enters Nakatomi Plaza and finds Holly's name on the computer so the guard can tell him where she's located in the building. After John does this, the guard tells him that she's on the 30th floor and that there's no one else left in the building. Now, why couldn't he have just told him that the party was on the 30th floor and saved him the trouble of playing with the computer?
Corrected entry: The black police officer keeps giving information about the police activities to McClane over the same frequency that he knows the terrorists are using. I also doubt he would be allowed to just use the police radio for chit-chat in such a situation.