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. (00:02:20)

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. (00:23:05 - 00:56:30)

Continuity mistake: When the terrorists launch the rocket at the RV, they break the same window of the building twice. (01:11:55)
Continuity mistake: When Hans takes Holly hostage, as she is sitting on the floor talking to Hans, as the camera goes from him to her, each time Holly's shirt is unbuttoned and open more so her bra and boobs are revealed.
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. (00:39:15)

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. (00:18:15)

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. (00:09:25)

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. (01:50:30)

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. This is because the ambulance was a late addition to the script - the terrorists leaving the truck was originally a longer shot, but had to be cut down as short as possible to hide the lack of ambulance. More info in the trivia section. (00:18:00)
Continuity mistake: When Bruce Willis shoots Alan Rickman, Rickman falls through the window and grabs Bonnie Bedelia'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. (01:57:15)
Other mistake: Powell says he has a desk job, but he is driving a cruiser. Cruisers are for cops on patrol. A cruiser would not be issued for a cop at a desk, especially not one who's on his way home, as Al says he is.
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.) (01:18:40)
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.
Other 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.
Revealing mistake: When John and the terrorist fall down the stairs together, you can hear the snap of the terrorist's neck signifying his death but at the bottom you can see the terrorist's "corpse" is still breathing. (00:36:00)
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.
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. (01:52:35)
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. (01:52:00)
Continuity mistake: When Bruce Willis breaks through the window (as the roof is blown), he lands on his back, head pointing towards the window. However the next moment we see him catching his breath, lying on his stomach and feet aimed at the window as the firehose hanger falls past the window. (01:52:10)
Character mistake: An unloaded automatic pistol is much lighter and balanced differently than a loaded one. When McClane hands "Bill Clay" (Gruber) the empty pistol, a terrorist of Gruber's credentials should pick up on the odd balance (I notice that difference, and I very rarely even pick up a gun).
Suggested correction: This presumes that a person can notice the difference between a fully-loaded pistol versus a pistol that is empty, which makes sense. However, it would be much more difficult for a person to tell that a gun is completely empty versus one that only has one or two bullets loaded. Since Gruber has no way of knowing his ruse didn't work, he doesn't believe McClane would have given him a completely empty gun. He obviously believes there is at the very least one bullet in the gun. He's not planning on getting into a shootout with McClane, he is just planning to threaten him and he certainly doesn't need a fully loaded gun to do that.
Suggested correction: It's easy to tell the different between the weight of a loaded vs. unloaded pistol, but given Hans had never held the pistol before, he had no way to compare the weight. Unless he regularly handled that particular model of pistol (unlikely, given his pistol is a completely different make and model) he would have no point of reference.
Most likely scenario. Different models will weigh differently as as stated, unless you got the point of reference how would you know?