Factual error: The Republic of Tunisia does not have mountains of the size seen when the hawk fights those who kidnapped the American soldier.
Other mistake: When walking through the Captain America display at the Smithsonian, Brody and Sam stand in front of a Captain America mannequin next to his shield. The mannequin is wearing boots of noticeably different heights.
Chosen answer: The sleeve probably would have been ripped to shreds in battle anyway. He was just saving time by ripping it himself. Also, gunfighters would take off their coats in a duel or an arm wrestler rolling up his sleeve.