When Saul was driving with Aileen, one shot shows that Aileen's car door was locked and the next shot, it was unlocked.
While Dana is arguing with her mother in the kitchen, the fall of her hair changes a number of times. It goes from falling over her right shoulder, to her left shoulder and evenly falling.
Marine One, the President's helicopter, does not land at Ronald Reagan National Airport, yet the roof of the house purchased by Aileen Morgan and her husband apparently has a direct line of sight to a helicopter pad reserved for that aircraft.
Immediately after being polygraphed, Brody asks if he passed and Carrie confidently says he did, "with flying colors." As any real CIA agent would know, polygraph results take weeks to calculate and assess. They aren't decided on the spot, and certainly not by an untrained observer.