When Carrie is introduced to Major Mendez the Major is seen wearing a Combat Infantrymen's Badge. Women are not eligible to serve in the Infantry and have never been authorized to wear this medal.
Dana is told that she needs to understand the "Quaker meeting" better. The Society of Friends is called "Quaker" colloquially; No Friend would ever refer formally to a meeting as a "Quaker" meeting.
Roya Hammad tells Brody he must steal an encryption key that unlocks a list of potential attack targets. However, encryption keys create secret messages, while decryption keys make those messages plaintext, so she should be saying decryption key.
When the Customs Officer is examining Carrie's Canadian passport upon landing in Lebanon, the area marked as "Signature of Bearer" is clearly unsigned. Canadians under 16 years of age are the only travelers exempt from signing their passports, thus Carrie's passport would not be valid for travel and the Customs Officer would have denied her entry.
When Saul is on the phone with Carrie, the background in his room shows a city. However, due to blowing wind and subsequent movement, this is revealed to be just a giant poster.
Carrie is shown entering Lebanon with a Canadian passport issued in 2006 with an expiry of 2016. Canada only started issuing passports with a 10-year expiry in 2013. Before 2013 all passports expired in 5 years.
While the Lebanese security chase after Saul (in about 9:07) there's a green sign in Hebrew denoting a garage ("shnap"). However, they speak Arabic in Beirut. The shooting location was in Jaffa, Israel.
When Carry is taken to CIA Nicosia, Cyprus' safe house, we see a house and at the background, sea.
The thing is Nicosia, the capital of Cyprus, has no sea.
The airport shown and quoted as "Beirut International" is in fact Dubai International Airport.