Continuity mistake: When the traitor first begins to shoot the 3 people, we see a close up on one of the victims (the one who gets shot in the head). If you look behind him in slow mo, you'll notice that the furniture is already red (from the blood).
Factual error: The scene where AF-1 is being refueled in the air is wrong. The common practice is that the tanker is flying in front of the other aircraft. They do not approach the plane from the back, and never that low. The flight path over the AF-1 by the KC-10 will create a turbulence which would be very dangerous if not fatal for the AF-1. The tanker will guide the plane towards the refueling boom. Also the guidelines to guide the refueling boom on the nose of the AF-1 are missing.
Continuity mistake: In at least one external shot with the plane on the ground, the wing tips have winglets (angled-up extensions of the wing) of a Boeing 747-400. In other shots, the winglets are missing.
Factual error: The two famous Air Force Ones are Boeing 747-200B model aircraft (military name is VC-25). That model has an older style flight deck, with steam/mechanical gauges. The newer 747-400 model has a "glass cockpit" with computer style displays. In the shot where the pilots get their takeoff clearance (and throughout the movie) you can see that it was filmed in a -400 model flight deck (most likely a training simulator for airlines).
Continuity mistake: In the beginning when the hijackers are trying to take off at the airport there is a shot where the plane goes over some emergency vehicles (before it takes off). These vehicles disappear in the following shot when the plane hits the tree branches. Also, a taxiing jet would've blown the vehicles and people around. Nothing of the sort happens in that scene.
Audio problem: When Ford and a terrorist are fighting in a side room, he knocks him out with the stool. He clearly grabs the stool by the metal legs, hitting him with the black cushion part and yet a metallic 'clang' can be heard. Even a metal bar against your head wouldn't make that noise.
Factual error: In the beginning scenes where the President's limousines are moving to the Moscow airport both cars have the US and Presidential flags on them. When travelling outside of the US the President's state cars fly the US flag and the flag of the visiting country. The Presidential flag is only flown with the US flag when the President is travelling in the United States.
Factual error: When the lead MiG-29 was about to fire his missile, before it was countermeasured by the flares, he flicks the A/G missile switch, which stands for Air-Ground, even though he had a full Air-air load.
Continuity mistake: When Agent Gibbs goes into the closed room and hands out the advance team reports from the Miami field agent, when he executes the three agents, if you watch, just after he kills the third agent, the agent's entire arm falls down behind the arm rest below the seat. However, in the following shots, when Gibbs re-holsters and turns to enter the armory, the agent's arm and hand are now risen upward, leaning against the window with his elbow propped up on the arm rest. (00:20:45)
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).