Revealing mistake: When John fights against the blonde German thug, Bruce Willis' stunt and the thug's are very noticeable: different body build plus darker and longer hair in the case of Willis, and bulky wig in the case of the German. (00:35:21)
Visible crew/equipment: When one of the terrorists enters an abandoned floor, logically the floor should be empty, but you can see the casting crew in the reflection of the windows. (00:35:00)
Revealing mistake: After escaping from the roof, John walks inside the building. When he runs towards a huge vent his bare feet swap to wearing black shoes.
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.
Revealing mistake: After barefoot John rolls down the stairs with the German thug one can notice the top of John's (his stunt in this case) protective sock around his left foot. (00:35:44)
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)
Visible crew/equipment: When John gets out of the limo at Nakatomi Plaza, a stage light is reflected clear as day in his door. (00:08:10)
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.
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.
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).
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.
Continuity mistake: In most of the film Bruce Willis' shoulder scar (from surgery when he was 17) is on his left, but as he is looking out the window just before SWAT comes in, it's on his right, clearly a flipped shot. (01:11:40)
Revealing mistake: The first time John walks up to the rooftop, he walks backwards to avoid the thugs' bullets and shelters behind a wall. Despite supposedly being barefoot his socks are very noticeable and wrinkle around the ankle.
Continuity mistake: Takagi leads John inside Holly's office. A shot later Takagi has moved several meters behind and is holding the door knob.