Continuity mistake: Bruce Willis is emerging from an underground air-vent when the plane with General Esperanza is landing. Initially the plane is coming on a direct course for him and he struggles to get out. In fact the front wheel is on course with and knocks the vent cover. But in the very next scene, the tire marks clearly show the front wheel passed to the right of the vent and so completely missed it.
Continuity mistake: When Col. Stuart is talking to Gen. Esperanza on the walkie-talkie about what runway to land on, behind Stuart you can see one of his men walk up to the map of the runways. Stuart turns around and starts walking to the map and the guy isn't there anymore. When Stuart gets in front of the map, the guy comes up and tells him which runway Esperanza should land on.
Continuity mistake: The dead bodies of the SWAT team move around on the escalator between shots. The best example of this is after the last SWAT guy dies, there is a body or two near the end of the escalator. They disappear or at least move much further away after McClane drops his gun on the escalator. And all this happens while the escalator is not moving.
Revealing mistake: Throughout the whole movie, you can see bits of 'snow' stuck to various things in close up view (i.e. guns, hair, faces, clothing). Look closely at the snow, its painfully obvious that they are little thin scraps of white plastic. Most evident is in McClane's hair when the platoon is about to move in on the church, the shot of him saying "You're not such an asshole, after all."
Continuity mistake: When McClane is desperately trying to get out of the manhole cover to avoid the plane, he props it open with a steel bar. Moments later, just before the plane runs over it, the bar is gone, so there should be nothing holding it open. The size of the gap between cover and runway also decreases between these shots.
Continuity mistake: John McClane and Major Grant are fighting on the wing of the aeroplane, and Major Grant's mouth gets extremely bloody. The shot switches to McClane and then back to Grant, whose mouth is still bloody. However, after the shot switches back from McClane to Grant, Grant's mouth is no longer bloody.
Continuity mistake: Two guys dressed in red go to the church and are let in by the old man. One of the men then says something like "You were right about that," and the scene shows him shoot his gun with his arm outstretched (obviously). The next scene though of the old man falling backwards shows the shooter's arm down by his side, however in the next scene where it cuts back to the shooter his arm is outstretched with the gun again.