When Alex Gardner and Dr. Novotny are sitting at the bar with the pitcher of beer, Novotny's glass goes from near empty to full, without being filled.
When Alex enters the volunteer's dream for the first time, he jumps/falls on the construction girder trying to rescue the man. While on the girder, Alex is alternatively in front on or behind the rope while reaching for the man (most noticeable where he uses his legs to try and anchor himself in place while reaching).
Blair may be the head of the CIA but he is not in charge of security for the President. That falls solely on the Secret Service for which Blair and Novotny , would have no say in the matter of where the President would be staying.
When Ms. Matusik has a heart attack, the paddles used to try and revive her are placed in the wrong spots on her chest. One paddle is supposed to be placed on the upper right of the patient and the other on the lower left at the base of the ribs. This allows for proper current travel through the heart. But both paddles are placed on her upper chest at her shoulders. This would do nothing for the patient. Also, you can see the actor jump before she is "shocked."
When Alex is collecting his money at the horse track he says he loves Ben Franklin because he discovered electricity. In truth, Franklin only determined that lighting was composed of electricity.
When Alex and Blair are sitting in the back of the moving car, you can see a moth fly by in the background. A car's aerodynamics wouldn't allow such an occurrence.
About an hour fifteen minutes in, Alex is trying to escape the bad guys out to kill him. At the racetrack, he rides the dirt bike down some stairs in what is obviously the seating area of the track. At the bottom of this public area a number of bales of wheat have been, luckily for Alex, placed. There is absolutely no logical reason why the bales would be there.
When Alex is playing his saxophone in his apartment near the beginning of the film, he is clearly not playing the instrument.
During the dream sequence at the end of the film, the camera crew is reflected in the wall as Blair rounds the corner to the elevator.
As Alex is sneaking back into the facility at the loading dock doors, 2 armed Government agents walk by on patrol with a dog (a Doberman Pinscher). Alex hides behind a nearby bush, while the agents and the dog walk past without noticing him. Obviously, the agents are using the dog to foil anyone simply 'hiding' from them, yet the dog does not notice Alex.