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."
Revealing mistake: When the army guy opens the door on the 747 to climb out onto the wing, the outside door handle does not move and looks painted on. When the second guy re-enters the plane and shuts the same door, there is no locking mechanism shown on the sides of the door; it simply pushes open.