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.
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.
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: 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.
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.
Other mistake: McClane's undershirt is white for the beginning of the movie, but when he emerges from the air conditioning ducts, it is now a dark colored shirt (it looks very dark green) with a lot of stains on it. If this is supposed to represent the same shirt, just very dirty, it could not have come from the A/C duct work, for it was all brand new and the interior shots showed it all to be spotlessly clean. The change in appearance is far too abrupt to be possible.