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.)
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.
Factual error: When Willis kills the terrorist and throws him out the window onto the police car, the body lands on the windshield, and it shatters into little bits. The windshields on 80's cars were double paned with a thin, strong plastic between each pane, to keep them from breaking into little bits in accidents. Only the side and back windows would shatter like that.
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.
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.
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.