Errors in geography
The time difference between Tehran and Washington, D.C., is 8.5 hours, and the difference between Washington and Los Aangeles is another three (i.e. 8:30 a.m. in Tehran is midnight in D.C, and 9:00 p.m. in L.A.). On the day of the escape, when Tony confirms to the CIA that he is going ahead with the mission, it appears to be dawn in Tehran and the scenes which follow over the next several hours are difficult to reconcile with these time zone differences.
It is stated that the British and New Zealand embassies refused to help staff from the American embassy. This was not true. Both the British and the New Zealand embassies sheltered the Americans, then helped to pass them on to the Canadians. Britain's Arthur Wyatt was later awarded the Companion of the Order of St. Michael and St. George for the risks that he took.
The opening narration states that, in 1953, the U.K. and the C.I.A. engineered a coup d'etat to depose Prime Minister Mohammmad Mossadegh and install Reza Pahlavi as the Shah. Mohammad Reza Pahlavi was already the Shah at the time of the coup. The coup began when the Shah dismissed Mossadegh and appointed Fazlollah Zahedi as the prime minister.
Incorrectly regarded as goofs
During the final sequence between Tony and his sleeping son Ian, a toy of the Star Wars character Boba Fett can be seen on the shelf. While many believe that the character of Boba Fett was not introduced until Star Wars: Episode V - The Empire Strikes Back (1980), he actually first appeared in The Star Wars Holiday Special (1978) (and had appeared in a parade a few weeks earlier still) and even the regular-sized figure originally came out in 1979 (as a special mail away offer).
When Sahar crosses the border into Iraq, the flag at the border station has "Allahu Akbar" in traditional kufic Arabic script between the three black stars. Saddam Hussein added this phrase in 1991 in what was supposed to be his handwriting. The kufic variant was added in 2004 following the American invasion. The flag in 1978 would have just displayed the stars.