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.
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: 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.
Factual error: The man who approaches Carrie in the bar identifies himself as a 'lieutenant JG', and the pilot who bombed the wedding party in the farmhouse. However, it was F-15s that dropped the bombs on the farmhouse, making them US Air Force aircraft. The USAF rank structure doesn't include the 'JG' rank- that's a US Navy rank.
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: The Pakistan Army uses British-style rank insignia with the crown replaced with the star and crescent of Pakistan. The ISI officers are all wearing another badge instead of the star and crescent. They are also wearing black gorget (collar) patches, whereas they should be wearing scarlet. Lt Col Khan and a number of other more junior officers should not be wearing gorget patches at all, as they are restricted to full colonels and above. ISI officers do not have a separate uniform; they wear ordinary military uniform if they are entitled to it.
Continuity mistake: Just before Carrie is told Brody is going to Bluemont Park, Carrie is shown re-watching Brody's welcoming ceremony on her laptop with the speech the Vice President gave and the speech Brody gave. The footage Carrie is watching on her laptop isn't the same as when the ceremony was broadcast earlier in the episode. The footage on the laptop has the Vice President saying "Sergeant Brody, you're a man who's endured a terrible ordeal" followed by Brody walking up the podium saying "Thank you, Mr. Vice President." But what was shown earlier was the Vice President welcoming Brody home, nothing along the lines of "Sergeant Brody, you're a man who's endured a terrible ordeal." Also, the length of time Brody spent walking to the podium and saying "Thank you, Mr. Vice President" has decreased when re-watching the ceremony on the radio - the actual ceremony showed Brody walking up to the podium for about 3-4 seconds followed by a 1-2 second delay before saying "Thank you, Mr. Vice President", but re-watching it on the laptop, Brody walks up to the podium and says "Thank you, Mr. Vice President" all within a split second of replying.
Factual error: Throughout the season, it is the BND who run security investigations within Germany. The BND is the German external intelligence service. The internal security service, who would actually be running these investigations, is the BfV. Neither service has a power of arrest, which BND agents exercise on at least two occasions during the season. If they need to arrest somebody then they must call in the Bundespolizei (as they do correctly at other times).
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).