Factual error: The passenger door at the front of a 747 opens outwards and towards the front of the plane (as seen when President Marshall boards the plane). When the first family is being evacuated at the end of the movie, the door is nowhere to be seen.
Suggested correction: That's because Gibbs had blown the emergency lock on the door and it went flying. Now, whether they'd actually do that and risk the door flying through the engine is another story, but I doubt the engineers had a mid-air rescue in mind when they designed it.
Continuity mistake: When Agent Gibbs shoots the 3 guards on the airplane, the last one who is shot keeps both the arms lowered. Then, in a next shot he is with the left arm lifted up. (00:20:35)
Continuity mistake: When Gary Oldman is boarding Air Force One, his ID shows him at 5'11" (which I thought was rather generous) and then later in the movie it's shown as 5'8."
Other mistake: When Gibbs goes into the Secret Service Room, he passes by an agent who he asks if he is sleeping on the job. He kills the other 3 agents, opens the weapons hold and walks past where the other agent was sitting, and he's now gone, and is still gone when the terrorists storm the cabin. Where did he go? Considering earlier during the tour, Melanie Mitchell warns them not to pass that point without an escort, or they'll be sorry.
Continuity mistake: When the President is finally discovered, they bring him upstairs. William H. Macy's hands are "taped behind his back" but notice when Ford finally overtakes Oldman, Macy's hands magically appear in front of him and he overpowers one of the guards. Finally he tells Ford to run after Oldman to stop the release, and his hand is taped behind his back again.
Character mistake: At the end of the movie, the rescue plane changes its callsign to Air Force One because the protocol is "Air Force One" is the callsign for any U.S. military plane (or, more specifically, an Air Force Plane) with the President on board, not just the blue and white 747. Thus, it would not be used for a plane on which the president is not on board. This is an important detail considering that for most of the middle section of the film, the terrorists believe the president has escaped and that they are dealing with a random secret service agent resisting them. The vice president and other administration officials dealing with the terrorists don't want them to know the president is still on board, as it could motivate the terrorists to threaten his family further. So, when the fuel tanker shows up to refuel the plane and addresses it as "Air Force One" to give instructions on the procedure, they are inadvertently confirming that the president is on board. To maintain the ruse, they should use the callsign "SAM-28000" or "Air Force 28000" when talking to the terrorists, referring to the plane's tail number. Similarly, any time an official makes a statement about the incident in public, they could refer to the plane as "28000" to keep up the ruse to the press (though it's not uncommon to refer to the 747s as "Air Force One" for the sake of simplicity in casual or non-official capacities, an instance of one plane communicating with another would not be).