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.
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.
Katie Holmes remarked that when her character,Sam,had to walk barefoot down a ladder,the rungs hurt her feet so badly that she was only able to do one take .And she never mentioned this to anybody on set?It wouldn't have been hard to have the prop department line the rungs of the ladder with padding.
When Mia's former roommate plays "Hail to the Chief" on her trombone, the slide movements do not match the notes she's playing.
Near the beginning, Samantha eats cake in the kitchen, and her father joins her. He asks her if she is ready, as they begin to dance. She answers that she is, yet her lips do not move.
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.
When James and Sam telling each other things they don't know, James says that every state in the union has a city named Springfield. This is not true, as Alaska, Arizona, Connecticut, Hawaii, Kansas, Montana, Nevada, New Mexico, North Dakota, Oklahoma, Rhode Island, Utah, Washington, and Wyoming are Springfield-less.
Spoiler: in the scenes with just James and Samantha, it is clear that he is a secret service agent as Samantha's other agents, Dylan and Bock, are not around. If he were not an agent, then the other secret service agents would have lost their jobs for losing Samantha on so many occasions and letting her go unprotected. The very first clue is when he leaves after giving Samantha the hat and sweatshirt. James leaves and then they are able to escape from the situation without any other agents on the floor. It is obvious that he told the other agents what was going on before they left.
After the day at the fair, when Samantha goes back to kiss James, she walks almost all the way to him while his back is to her. Then she says his name, presumably to get him to turn around, but then it is seen that he is already facing her, so there was no reason for her say "James" in the way she did.