Factual error: It's mentioned that the British and New Zealanders turned the Americans away. This is untrue - the Americans stayed with both for a time until it was decided that they would be safer with the Canadians.
Continuity mistake: Ben Affleck gets into a 280s Mercedes driven by the Canadian Ambassador. The front screen wipers on the car are original silver chrome style and fit the screen. When they arrive at his house the wipers have now changed to much shorter modern style wipers, are black, and are the very latest frameless style first introduced in the early 2000's by Bosch.
Plot hole: Just after the scene in which the Embassy gates crash open allowing floods of Iranians in, the Embassy shreds/incinerates as many sensitive files as they can. In the scene where the power goes out, as well as the incinerator, the two characters decide the remaining trolley load of files should be shredded with a shredder, which needs electricity to work.
Factual error: In the scene where the Swissair 747 is taking off from Tehran the police cars keep up with the plane until it lifts off. Takeoff speed for a 747 is about 160 mph so the police cars would have been far behind by then.
Factual error: They play a track "When the Levee Breaks" from a Led Zeppelin album. The hand shows the needle being placed at the next to last track on the album. However, "When the Levee breaks" is the last track on Led Zeppelin IV's second side.
Factual error: The glasses, which John Goodman is wearing throughout this movie are called Ray Ban Clubmaster. They premiered in 1986, but the movie is set in 1980.
Continuity mistake: There is a scene after Tony came up with the idea for the fake movie and presented it to the CIA, where he is looking at papers under a magnifying glass. Tony is holding his cigarette in his left hand under the magnifying glass. The very next shot shows his face, and he suddenly has the cigarette in his mouth. Which he then removes with his right hand. The next shot shows it in his left hand again.
Factual error: Iranian revolutionary guards didn't have caps with their uniforms until 1988 (after the end of the Iran-Iraq war).
Factual error: During the first movie lot scene, John Goodman walks past a 1990's Ez-Go golf cart. Another is seen in the background.
Factual error: The embassy takeover was November 3rd. A scene in Washington DC described as 69 days later showed Tony Mendez driving around town with yellow/fall like leaves on the trees, something that would not have been true in the middle of January. A similar scene at the end of the movie has Tony Mendez approaching his house in Virginia and a very fall like scene is shown.
Factual error: Early in the movie in the American Embassy, a reprint of a WPA poster of Grand Canyon is seen on the wall behind a desk. Those reprints were not started until the late 1980s.
Continuity mistake: When the State Department approves the exfiltration, Ben Affleck is dropped at Dulles Airport (it has a distinctive swooping roof), but after he boards, the plane is seen taking off from Reagan National airport. The evidence is the fact that you see a bridge (the Woodrow Wilson bridge, which was rebuilt in 2006 and can be seen in its current form) in the background.
Factual error: In 1980 the Canadian airport we see was called Dorval international airport, not Trudeau.
Factual error: The turbans worn in the scenes in Tehran are not the sort worn by Iranian clergy - they are wound incorrectly and are too flat.
Factual error: At the press conference for the bogus movie Argo the actor dressed as a C3P0-knockoff robot has blue-white LEDs for eyes in his costume - which were not available until well into the 1990s.
Factual error: During the protests in the opening scene protesters are chanting "ubij ga", meaning "kill him" but in Serbo-Croatian or Bosnian language, definitely not in any Iranian language.
Factual error: The Swiss Air 747 in the movie had an extended upper deck. The first Swiss Air 747-300 did not fly until 1982, and didn't enter commercial service until 1983, well after the events of this movie.
Factual error: In the beginning of the film, when Agent Mendez is called to CIA Headquarters, as he passes the secretary's desk, the outgoing letter propped against her word processor has two postage stamps affixed (2-cent Navajo necklace & 37-cent US Flag) which were first issued and then reissued by the US Postal Service beginning 2003-04, 23+ years after the hostage event in 1980.
Factual error: CIA agent Mendez takes-off from Washington to Tehran, apparently via London because he boarded a British Airways flight, on a twin engine aircraft. Twin engine aircraft flights across the Atlantic were just being introduced by TWA around 1980 and weren't flown by BA until years later, if at all. Also, the BA Boeing 747 depicted had tail colors that were not introduced by the airline until the early 2000s.