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.
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.
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.
Other mistake: Brody is getting his hair cut, a man enters the room and says "We're ready for you Sergeant Brody, your debriefing." Brody replies "I want to call my wife first." It then cuts to Mike and Jessica. Brody calls Jessica answers, she gets dressed then leaves to go home. Brody didn't call the family residence however, he called Mike's house. How would he have known she was there? And why wouldn't he call her at home?
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).