At the end of the movie, Fischer is shown getting into a car and pulling his portable chess-set from his pocket. The scene then changes to a close-up of his hands which open the set vertically with the inner flap folding downward. The shot switches back to a facing shot of Fischer who is now holding the set opened horizontally, like a book with the flap opened to his right. Then we once again see a close-up of his hands folding the set closed vertically.
When Fischer first meets Boris Spassky in Santa Monica in 1966 Boris is introduced as the World Champion. Boris was not yet the World Champion. Tigran Petrosian was the World Chess Champion at that time.
Fischer is shown resigning the first game immediately after Spassky traps his bishop. In fact, he played on for another 26 moves before giving up.
Paul Morphy didn't commit suicide; he died of a stroke at age 47.
While in real life the Soviet delegation did ask for Spassky's chair to be examined for electronic devices, that did NOT happen during a game. In the film, Spassky overturned the chair himself to examine it while the game was going on. Such clownish behavior would have caused an uproar from the audience - not to mention broken numerous match rules (disturbing his opponent, etc.)
While Fischer's distrust of the Soviets was well-known, at the time of the 1972 Championship Match, there was no evidence or even suggestion that he had mental disorder or that he was attracted to anti-Semitic propaganda or literature. All those came a decade or more later, when he was no longer World Chess Champion.
Near the start, when young Bobby is looking out the window of the Fischer family's front door, the date on the screen reads 15 Nov. 1951. Bobby goes to see his mother and tells her there's a red 1952 Chevy outside. In the American car market, model years typically begin in the last quarter of the previous year, so Bobby could have seen a '52 Chevy in late '51.
The first move young Bobby Fischer visualizes on the chessboard is labeled as "P-B4" although the pawn is actually moving on the C file. In algebraic chess notation, the move would be given as "c4". However, the older descriptive notation was still standard in English-speaking countries in the 1950s, and Bobby used it his entire career. The move, and all the other moves Bobby visualizes, are given correctly in descriptive notation.
On the first chess board young Bobby Fischer is analyzing, he chooses to make a "pawn sacrifice", but there is a hanging rook on the back rank. The move makes no sense.
Title card on screen announces 1965 for a scene where Paul Marshall hands Fischer his card, boasting of what he did promoting Jimi Hendrix and also the Rolling Stones. In 1965 Hendrix was still an unknown, itinerant musician who would not form his successful band The Jimi Hendrix Experience in England until Fall 1966, so the name-dropping in this line of dialog is premature by about two or three years.
The lawyer Marshall says in a scene with the priest that "for him (Fisher) Vietnam and the Beatles never happened" This is said in Jan 1965. US troops did not start fighting in Vietnam until the Battle of Ia Drang in November 1965 so calling it "Vietnam" which was a general term later used for the war in the late 60's to today would not have had much meaning since it was not a big issue yet and the US would not even start military bombing until Feb 1965.
In several close ups, contact lenses are visible in Tobey Maguire's eyes.
The auditorium's emergency exit signs in multiple shots are modern designs not utilized in 1972 when the film takes place.