When Bruce is looking on the touch-screen computer to find out where his wife is, he touches the name GENNARO, but when it changes colour it also corrects the spelling of his wife's name to GENNERO.
When the terrorists enter the building, they stream out of the rental truck. You don't see the large ambulance that emerges for their getaway at the end of the film.
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) - before the camera cuts out he realises he's about to hit the doorframe very hard. Reminiscent of the Stormtrooper in Star Wars.
Bruce Willis wore fake feet during the movie to protect his own while shooting the film. They were like boots that slipped on over his. At the part where the FBI guy shoots at him from the helicopter and Bruce jumps to a lower level of the roof, you can see the tops of the fake feet that aren't tight around his ankles.
See all the mistakes in Die Hard