Factual error: In the scene where AF1 gets refueled by a KC-10 Extender tanker, it would be impossible to pull the whole refueling probe out of the tanker plane, not to mention causing a fire. AF1 would simply break the end of the probe, the tanker plane crew would close the internal fuel valves, and there would be no fire. The refueling hardware used on USAF tanker aircraft has predetermined breaking points, since it's likely that any other aircraft getting refueled suddenly loses height, gets off course or something.
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.
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).
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.