When Samantha takes Mia and James to the dance, right after President Mackenzie says he wants to dance with his wife, the band starts playing. The trumpets are heard to be playing with Harmon mutes, however, the trumpets sitting on stage have no mutes in.
During the scene with James and Sam telling each other things they don't know, James says that every state in the union has a city named Springfield; not true, Alaska, Arizona, Connecticut, Hawaii, Iowa, Kansas, Montana, Nevada, New Mexico, North Dakota, Oklahoma, Rhode Island, Utah, Washington, and Wyoming are all Springfield-less.
The paint job on Air Force One is dull and poorly colored. The real Air Force One has a shiny bright blue top, a light blue forward lower portion and the aft bottom is highly polished aluminum. This aircraft is dull and non-reflective.
The interior of Air Force One is shown to have loose glasses, vases, etc., sitting on the walls and on tables. All of those items would be secured in an actual aircraft, otherwise they would fall to floor as the aircraft rotates, or if it hit any amount of turbulence.