Other mistake: When John is on the roof being shot at, he shoots the lock off a door before charging through it. We see a close up of the door as it's shot, and there's a blood smear already on it. He then goes through it shoulder first, which is what would leave a smear like that - must be from an earlier take.
Continuity mistake: The scene after John jumps from the roof and enters the building, he jumps in a pond of water to evade the helicopter blast and comes out completely drenched, but in the next scene he is completely dry.
Continuity mistake: After John has crawled through the vents and enters the room where Hans shot Takagi, the amount of blood on the window is a lot more than was thereafter Takagi got shot.
Continuity mistake: The headlights on Al's cop car are off in McClane's view from above but back on from ground level.
Other mistake: All of the lights on one of the Christmas trees stay on throughout the movie, even though the power was cut, and only emergency lighting supposedly was left on.
Visible crew/equipment: Just after John has shot Hans, he falls and smashes the glass window. Look behind him: you can see a wire in two shots. It's meant to hold the stunt double for either Holly or Hans. No slow-mo required.
Continuity mistake: When the box truck (carrying the thief gang) drives down the ramp and the grey Mercedes continues on, you can see that the driver of the Mercedes is a white male without glasses. A minute later, the grey Mercedes pulls up to the front of Nakatomi Plaza. Alexander Gudunov gets out on the passenger side, and the driver gets out: he's now a black male with glasses.
Other mistake: When the terrorists shoot the armoured vehicle from high up in the building, the angle at which the rocket launcher is fired seen from inside the building is simply wrong for a target that is virtually directly below them. The angle shown would have hit much further away. This is true for both shots but more obvious for the second one.
Factual error: When the police officer first comes to the tower and looks around there is a football game the terrorist is watching on television. The movie takes place on Christmas Eve. The Southern California vs. Notre Dame game, which is the one being played in the movie, is a classic rivalry game played in either late November or early October. This is especially true in 1988 as ND and USC were ranked 1 and 2. Notice the terrorist display displeasure when the announcer notes that ND just scored. USC is the home town team. It is also apparent that the game is being played during daylight hours, but at this point it is dark outside.
Factual error: If the hose hanger isn't heavy enough to instantly yank McClane out of the window, but instead slowly pulls him down, then it should be light enough for McClane to simply stand up and reel it in. If his weight is enough to slow the hanger's decent, then that means it only weighs slightly more than he does. He should have no trouble standing up and walking backwards to pull the hanger back up. The mistake is that the hanger pulls him slowly at all, it should have yanked him directly out of the window the second the line ran out of slack because McClane is not anchored to the floor in any way. This scene suggest that the hanger is both heavy enough to pull McClane but light enough that McClane's weight significantly slows it, which is impossible.
Continuity mistake: During the fight between Karl and McClane, Karl shoots at John and it hits him in the shoulder. That wound then disappears and reappears several times throughout the movie.
Revealing mistake: Having leapt from the rooftop, Bruce has his feet planted on the window. As he then goes to push himself back in order to smash the glass with his gun shots watch the rope. Someone above has kindly held the rope out several meters away from the building in the next shot, allowing him to get a much bigger swing.
Factual error: Cutting through phone lines will not electrocute you. At best there a 80v AC ringing current per line which gives a mild shock. Also, when the technical expert kits the data cabinets the sensitive electrical equipment is highly unlikely to explode in a shower of sparks.
Continuity mistake: When John and Al are talking, if they have any kids, there is a shot of John's family in his wallet and you can see both thumbs on the wallet. The scene cuts away to John talking, and his left hand is holding the walkie-talkie. There isn't enough time for John to grab the walkie and start talking into it.
Factual error: Obviously done to spice up the action, but there would in no way be enough time for McClane to jump away when he sees the rising inferno of flames coming up the elevator, as it would be too fast and burn him to a crisp.
Revealing mistake: As Bruce Willis walks into Nakatomi Tower, which is solely occupied by the Nakatomi corporation, behind him can be seen the logo of Merryl Lynch, which had offices in the building used for filming.
Factual error: Would pulling the alarm call the fire and police departments if the phone lines were disabled?
Suggested correction: While I'm not sure how the alarms contact the fire department, it's also important to note that when Karl cuts the phone lines, his brother is re-routing some of the lines. I believe the implication is that he was bypassing certain lines so that way emergency services and the phone company wouldn't get alerts that their phones lines were down at the plaza and send people to check it out. So it would makes sense to me that a fire alarm could still get a signal through the phone lines in that case. (I'm also presuming that the way the fire alarm sends its signal is different from how a phone call is made, since one is automated and not call-based).
Continuity mistake: When the FBI shows up and Duane Robinson is explaining to them what the police know, Al tells the FBI about John. Duane is in front of Al in the shot looking at the building, and right next to him in the opposite angle.
Deliberate mistake: For Hans Gruber's iconic death scene, Alan Rickman's fall was filmed at high speed (for slow-motion playback) against a green screen, and the skyscraper perspective footage was added later as background. However, while Rickman falls away from the camera in slow motion, papers are fluttering around him in the background at normal speed. This was done deliberately to make the shot even more surreal.
Visible crew/equipment: The shadow of the camera is visible right of screen when John is wrapping himself with the fire hose and asking himself why he's doing it. It lingers for some time. It looks to be a steadycam with a monitor.
