Continuity mistake: When Forrest was running in the first football scene, it shows cheerleaders and a couple of football players on the sidelines, then the next scene in closer view of Forrest running it shows the same cheerleaders and players in the same exact spot as if Forrest ran that same part of the field again. (00:22:15)
Continuity mistake: When Forrest gets hit by a stone and falls on the ground a red school book is lying on the ground next to him. When he gets up he doesn't pick it up, and when he runs away it's not on the ground either. (00:15:20)
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).
Plot hole: When Forrest finds out he is receiving the Medal of Honor, he is addressed as PFC Gump (Private First Class, E3). "Two weeks later", he leaves Vietnam, and is then shown receiving the medal from President Johnson. In this scene, he is wearing Sergeant's stripes (E5). There is no indication or mention of him receiving a promotion, let alone two. (01:02:10 - 01:03:05)