The helicopter that Bishop is flying in to the White House changes from a UH-1 Huey to the President's helicopter Marine One in the shot involving the north side of the White House, and then changes back to the UH-1 Huey when it lands on the south lawn.
In the beginning of the movie when Bishop gets dressed to meet Congressman Page and later to conduct the news conference he is wearing a dark brown two-piece suit but when he show up at dinner with Conrad, Bishop is wearing a three-piece dark blue suit.
The final scene involves Sheen throwing a basketball with great force to a basketball hoop (revealed a little later) that is a considerable distance away. The ball bounces lamely (in slow motion) on the ring/hoop - twice - but it should have bounced off with a force related to the supposed speed of its arrival.
In the beginning when the clean-up-crew arrive you can clearly see one of the "dead" professor breathing.
When Bishop tries to get files on some of the "Shadow" members, "Joint Chief of Staff" is misspelled "Chiief" at the bottom of the monitor.
The satellite zoom-in near the beginning of the film first shows Albemarle Sound in North Carolina, then zooms in to the Baltimore, Maryland harbor, several hundred miles north of the first image.