Continuity mistake: When Takagi explains to the terrorists that, "he doesn't have that code," the trees outside the window are clearly swaying in the wind. The next shot is a panoramic view of the room, but now, the trees outside are still. (00:27:50)
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.
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.
Factual error: The shrink on the news refers to the Helsinki syndrome. This should be the Stockholm syndrome. Being an expert, he should know the difference, and there's no suggestion it's a deliberate oversight to imply a lack of knowledge. (01:25:25)
Factual error: The walkie talkies used are Kenwood TH21BT 2 meter VHF ham radio walkie talkies. Stubby rubber ducky antennas, if they are made for VHF, wouldn't transmit very far - likely they are UHF antennas which is even worse. The police dispatchers would not be listening to them on any Channel 9 on 2 meter or 440 Mhz ham radio. Also Al Powell would not have a radio for that in his car. If there was a Channel 9, an emergency channel, Hans Gruber's radio would not have been tuned to that during John McClane's rooftop call for help picked up by police - monitoring channel 9? Also on Thornberg's desk at the TV station there is a Kenwood TS-711 according to RigPix Database info (I thought it was an HF rig) - with the proper antenna it would allow Thornberg to communicate from his office with John McClane and the terrorists. But why would he need to do that? He doesn't seem to be the kind of guy to have an All Mode (FM, SSB, CW) VHF transceiver on his desktop. Probably most hams don't even have that. Maybe an FM 2 meter and 440, maybe an HF or HF and 6 Meters. But only a VHF all mode (expensive for single purpose) interested ham radio operator would have such a rig.
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: John McClane wraps a fire hose around his waist, uses it to jump 100 feet until the coil catches the lip of the building, and the hose catches him. This should have seriously injured him due to the inelastic material fire hoses are made of, and the fact that he was not wearing a safety harness, but he barely gets the wind knocked out of him.
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.
Continuity mistake: When John and Karl are fighting and Karl reaches for the gun lying on the floor, John legs it out of an open door. Karl fires two shots, the first hits John and the second sprays blood all over the door. (01:47:35)
Continuity mistake: After McClane "kills" Karl by hanging him with the chain, Karl can be spotted at the top of the stairs on the right side as McClane is reloading his gun. (01:53:30)
Character mistake: In the infamous 'shoot the glass' scene, Alan Rickman, playing a German, commands to 'Schieß den Fenster', which is wrong in German because the 'Fenster', the window, has in fact the neutral article 'das' instead of the masculine 'den'. Also, it is not a window in German but a 'Fensterscheibe', a window pane. In German he would say 'Schieß auf das Fenster' ('Shoot at the window!') or 'Schieß auf die Scheiben' ('Shoot the window panes!'). (01:36:00 - 01:37:30)