During the courtroom scene, 21 minutes and 5 seconds into the film, Thompson sets down his glass which is almost full. A second later, a close-up on the glass shows it sitting on the floor, empty. All further scenes in the courtroom show the glass once again almost full.
In the bathroom scene when Murray is washing his shoes out in the sink, then hits them repeatedly against the urinal to get the excess water out, he is wearing glasses but his reflection in the mirror behind the urinals isn't.
When Dr. Thompson goes to college to "lecture" the students, he lights up a joint, takes a few hits off of it and then moments later lights up yet another joint before finishing the first one (or passing the first one along to anyone else) basically showing that he lit the same joint twice.
In 1972 Super Bowl VI was played in New Orleans at Tulane Stadium, not in Los Angeles, as depicted in the film.
On the closing credits, Creedence Clearwater Revival is misspelled as "Credence."
A number of cars in the hotel parking lot at the Super Bowl were models that didn't exist in 1972.
The plane of journalists following the 1972 campaign sports an Evergreen International label. Evergreen International Airlines was a cargo airline formed in November 1975.
(at around 1h 10 mins) When Thompson addresses the college students in the auditorium, he lights a joint with a Bic lighter. The scene took place in 1972, and the Bic disposable lighter wasn't invented until 1973.