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.
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.
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.
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.
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.
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.