Continuity mistake: When the terrorists launch the rocket at the RV, they break the same window of the building twice.
Revealing mistake: When Hans discovers the first bad guy lying dead in the lift ("Now I have a gun - HO HO HO!") and pushes his head, the corpse is moving his head before he is hit (watch carefully). He also closes his eyes before Hans makes contact.
Other mistake: In the shot where we see all the terrorists (except Karl and Theo, who are upstairs) walking into the building from downstairs, they are about to walk through a doorway when the camera cuts out. Watch the terrorist on the left (he's the one who guards the door) - the way he's walking he's going to smack straight into the wall! The shot cuts a fraction of a second before he does, but that's not a remotely normal walk/position - it's clearly just because it makes for a cool looking shot.
Other mistake: When Bruce is looking on the touch-screen computer to find out where his wife is, he touches the name GENNARO, but when it changes colour it also corrects the spelling of his wife's name to GENNERO.
Continuity mistake: When the terrorists enter the building, they stream out of the rental truck. You don't see the large ambulance that emerges for their getaway at the end of the film.
Continuity mistake: When Bruce Willis shoots Alan Rickman, Rickman falls through the window and grabs Bonnie Bedilia's arm to pull her with him. He has a good grasp on her arm because of her watch. When Bruce Willis is trying to release the clasp on her watch, you see blood on her face from his arm. The shot then cuts to Alan Rickman, and when the camera goes back to Bedelia and Willis, there is no longer any blood on her face.
Revealing mistake: Bruce Willis wore fake feet during the movie to protect his own while shooting the film. They were like boots that slipped on over his. At the part where the FBI guy shoots at him from the helicopter and Bruce jumps to a lower level of the roof, you can see the tops of the fake feet that aren't tight around his ankles.
Continuity mistake: After he climbs out of the air duct, John McClane's shirt goes from white to green. The dirt and grime from the air duct would cause discolouration only where the shirt comes in contact with the duct. However, his shirt is uniformly green including places that wouldn't make contact and it is impossible for all surfaces to make contact even in small areas.
Other mistake: In the computer room shoot-out the bad guys shoot out all the glass room dividers in a hail of bullets, but the glass windows on the exterior of the building (behind the dividers and in the path of the bullets) stay intact; they don't even get scratched.
Factual error: Throughout the film, characters are able to break through on radio transmissions and speak to the other person talking, and the other person would hear them. In reality, they would have to wait until the first transmitter un-keyed the radio before any other discernable radio traffic could have be heard. (A good example of this is the scene where Ellis is giving up McClane.)
Visible crew/equipment: In the first scene of the Christmas party when Takagi steps out of one the offices and greets a couple of his guests, the camera pans left to show the orchestra and other guests. A huge shadow of the camera and its operator is being shown on the back wall, then the stage light gets moved to light a different area and the shadow is gone.
Continuity mistake: After the terrorists blow the roof and Willis jumps off with the fire hose, he hits the window. After kicking the window for a while, he decides to shoot it. As he pushes off the window you can see that his gun is out and pointed at the window yet in the next scene he pulls the gun from his side and then starts shooting at the window.
Other mistake: Near the start of the film, when John goes to meet his wife at the office block, there is a party going on and a small orchestra playing. When Gruber and his gang enter, the orchestra are still there. Gruber locks down the building, and takes the guests hostage. Where does the orchestra go after that? They are never seen again, even when the hostages are taken to the roof.
Revealing mistake: When the hostages are running down the stairs after the C4 explosion, a large, fist sized chunk of concrete falls on an older man's head and he just runs straight on.
Revealing mistake: After Powell frantically radios in his distress call, other police cars converge on the street in front of Nakatomi Plaza, blocking off both sides of the street. But when McClane grapples down the building using the firehose, you can see traffic on the street below and none of them are police.
Continuity mistake: Takagi is sitting on the chair facing the door of the room when he is shot. In the next scene, the same door is splattered with blood, yet the glass structure which should be splattered with blood should be the glass window behind Takagi, not the door which he is facing.
Factual error: The rocket-launcher used to destroy the LAPD Armoured car appears to be a French 'MISTRAL' infra-red homing, man-portable surface-to-air missile that is meant for shooting down aircraft and is not designed for a surface target role. Even if the missile seeker could acquire a surface target like the armored car as a correct infra-red signature (required before it will fire), its flight profile would not be suitable for attack and it would probably not even arm the missile warhead at such a close distance. The missile is also not re-loadable in the manner shown.