Revealing mistake: In the scene where Lt. Dan chases the hookers out of his apartment, if you look closely at how he's carrying himself you can see Sinise use his real leg (which was then hidden by the special effects people) to push himself back into his wheelchair.
Revealing mistake: In the black-and-white newsreel footage of Forrest's All American team visiting President Kennedy in the White House, Kennedy is handling a football and you can see to the right of his grip "N STATE" printed on the ball and beneath that the signature "Rip Engle." The producers obviously spliced stock footage of when the Penn State team visited the White House following their Gator Bowl win in 1962. Rip Engle was their coach at the time. (Kennedy's hand hides the first three letters of Penn State).
Revealing mistake: When Forrest is playing ping pong in the gym and an officer comes to give him his discharge papers, Forrest reaches for his paddle and the ball but when he picks up his hand, there is no ball although he is acting as if there were. The ping pong scenes used CGI to put the balls into the scenes and this was obviously overlooked.