When Nixon signs his resignation letter, two very different signatures are used.
The film shows Nixon signing his resignation letter the day before he leaves office and prior to it being publicly announced. Historically, Nixon informed the nation in an address the night before leaving office, and then signed the letter the next day, which was his last morning in the White House.
Lyndon Johnson's televised speech on his decision to not seek reelection is presented out of order. In the original speech, Johnson first talks about not accepting the Presidential nomination one more time ("Accordingly, I shall not seek and I will not accept" part) then proceeds to urge Americans in the fight of an honored cause ("whatever the price, whatever the burden" part). Oliver Stone obviously switched the order for dramatic effects while presenting another Johnson speech.
Nixon is shown awarding the Medal of Honor to an unnamed Naval Aviator who had lost his legs in the Vietnam War. In reality, only two Naval Aviators were awarded the Medal of Honor during the Vietnam War, VADM James B. Stockdale, and CDR Clyde E. Lassen, neither of whom had lost their legs.
The reunion between Nixon and Hoover is situated at Santa Anita Racetrack. In fact, as Clint Eastwood's "J. Edgar" rightly spots (though some goofs of its own), J. Edgar Hoover used to go to Del Mar Turf Club, a few miles from Scripps Clinic in La Jolla, where he made a medical checkup every year.
The incident where Nixon angrily pushed Press Secretary Ron Ziegler in front of a crowd of journalists didn't happened inside the White House during a press conference on the "peace with honor" agreement. The real incident took place during a Veterans of Foreign Wars convention in 1973.
Flipped shot: when Nixon enters the Beverly Hilton ballroom, the campaign signs are backwards.
Numerous scenes throughout the movie feature President Nixon seated or standing in front of a crackling log fire, particularly in the scene where he talks to the portrait of Abraham Lincoln. Although in each scene real logs can be heard crackling, hissing, and popping, as real wood will do, close-ups of the fire reveal it to be a "fake" natural gas fireplace, with artificial logs that are "burning" evenly and cleanly with a vertical flame, and no smoke or embers coming off them.
The logo on the Shure SM57 Microphones on the presidential podium has the newer style (post 1985) block lettering. They should have the older style Shure Unidyne III SM57 Cardioid Dynamic Microphone logo.
By the end of the movie, as Nixon talks with Kissinger after praying he pulls out his pen to sign his resignation letter. Clearly the pen he uses is a modern Parker Sonnet. This model was not to be produced before years later, circa 1993. Actually Nixon would've used the Nixon't pen of choice during his presidency a Parker 75 in plain sterling silver called Keepsake.
The road Nixon is traveling along to get to the ranch in 1972 has yellow markings. At that time, although the USA was replacing the white ones, country roads would have not been changed, and even if it had been changed, the marking would not have been faded.
In the scene in which Nixon is hospitalized with pneumonia, which happened in 1973, he hallucinates that he is lying in bed surrounded by tapes. The tapes shown are Ampex type 478, which was not introduced until 1988.
At about one hour 40 minutes in, when Nixon and his associates are discussing what to do about Daniel Ellsberg's leak of the Pentagon Papers and Nixon says to Kissinger "...this Ellsberg, wasn't he a student of yours at Harvard?" a long-haired crew member can be seen attempting to duck out of the right-hand side of the next shot.
During a scene when Nixon greeted military veteran, an African-American Marine General can be seen wearing four-star insignia on his shoulder. In the real life, as of August 2022 there's never been an African-American Marine Four-Star General until Lieutenant General Michael E. Langley was confirmed for his post as commander of USAFRICOM. A post which was commanded by Four-Star General and making Michael E. Langley the first African-American Marine Four-Star General in history.