Visible crew/equipment: In the first scene of the Christmas party when Takagi steps out of one the offices and greets a couple of his guests, the camera pans left to show the orchestra and other guests. A huge shadow of the camera and its operator is being shown on the back wall, then the stage light gets moved to light a different area and the shadow is gone.
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.
Question: Why didn't Hans Gruber simply place 5 hostages in a room and threaten to blow their brains out if John McClane doesn't hand himself in? John McClane is the good guy with a conscience and Hans Gruber is the ruthless killer that kills 2 people in a heartbeat, John would have been forced to hand himself in or be responsible for their deaths. Even if Hans didn't want to kill anyone, he could have pretended to shoot people one by one. John wouldn't know any better.
Question: On the some versions (TV), when Al walks in the lobby to check it out, right before he gets to the elevators and then leaves, the camera pans to the left and you see a terrorist with a machine gun waiting for him. On other versions (VHS) you don't see this terrorist. Why? NOTE: I've seen both on the same TV set.