Factual error: The Pakistan Air Force officer in the airport is listed in the credits as 'PAF Lieutenant' and is wearing the two pips of an army lieutenant. In fact, the PAF still uses RAF-style ranks, so he should be a Flying Officer, and the PAF also uses distinctive eight-pointed stars instead of pips.
Continuity mistake: When Sgt Brody first returns to the US, he's seen in uniform wearing a sergeant's three chevrons with one hash mark, indicating at least 4 years of service. Later, he's a Gunnery Sergeant, with two rockers below the chevrons, plus two additional hash marks to reflect the eight years he was a POW. In the episode "Achilles Heel," Brody and his wife attend a swank Capitol Hill party, but his dress blues have only the three stripes and single hash mark. Throughout the series, everyone addresses Brody as "Sergeant" and never as "Gunny," which is the traditional reference for a Gunnery Sergeant. (00:22:00 - 00:28:00)
Continuity mistake: Quinn is clean shaven when sitting by the pool with his girlfriend. The action shifts to Carrie in Pakistan and then back to DC where Dar confronts Quinn in the motel. Quinn now has a mustache and a bit of goatee, which he has for the balance of the episode. While the precise timeline is unclear, everything is fast paced and it seems that virtually no time has elapsed, certainly not enough time for Quinn to have grown a mustache.
Plot hole: Later in the evening, after NYPD takes Quinn into custody, Carrie finds his cell phone, with pictures showing their home had been spied upon from across the street, as well as other evidence of a conspiracy. He had tried to hand her the phone right after the police broke in, but it was knocked from his hand and she discovered it under the smashed door. In the real world, Carrie's home would have been considered a crime scene and the police would have searched it thoroughly, top to bottom, undoubtedly finding the phone themselves, but are shown driving away after the confrontation.
Factual error: As Brody is driven away from the square after meeting Abu Nazir's widow, the shot from inside his car reveals a public telephone box on the corner. He is supposed to be in Tehran, Iran - however the distinctive colours of the phone box are those of Morocco (where I assume the scene was shot). (00:35:40)