Deliberate mistake: It's not particularly believable that the away team to stop the assassination would consist of the ship's entire senior staff, and raises the question of who was left in charge of the bridge (the answer: apparently nobody, as they return to a completely deserted bridge). Of course this is dramatic license, and at this point the audience won't care or likely even notice.
Deliberate mistake: The eyewitness testimony that placed Oswald in the lunchroom at the book despository didn't come to light until the House Select Committee on Assassination in 1978, many years after the Shaw trial.