When Forrest leaves the bank after trying to pay off Jewel's mortgage, he is seen in an overhead shot. Truncated domes are visible on the sidewalk near the street crossing (the 2 yellow panels) The movie takes place in the early 1980s, and truncated domes were not required in the United States until the Americans with Disabilities Act of 1991. They were practically non existent in the U.S. before then.
During the opening scenes while Forrest is eluding the police, the radio dispatcher calls out that he's driving a white 4 door sedan and then a blue 4 door sedan when in fact both cars are clearly 2 door coupes.
During the research scene prior to the bank heist on the roof the armoured car is number #5012. Years later after his prison sentence an armoured car goes past as Forrest and Jewel walk down the street. The exact same armoured car is used ..... also numbered #5012.
During the ride with Spacek after picking her up from her broken truck, the door lock is in the locked position, then unlocked, then locked, and is unlocked every time we see her at the end of that driving scene.
Besides using the usual movie convention of not using local TV channel numbers for newscasts, the news stories displayed were typical of smaller town station packages and looked more like early seventies news. Dallas TV news departments KDFW, KXAS, KTVT and especially WFAA-TV were high quality operations and WFAA's 1981 operation included many specialized "beat" reporters focusing on detailed investigation of business, crime, medicine, politics and sports.
A pensive Jewel (Sissy Spacek) almost overfills a kettle with water. She then pours some water out and puts the kettle on the stove. She walks around for a bit, and mere seconds later, in the same take, the kettle starts whistling off screen. The amount of water in it would have taken several minutes to even warm up, let alone boil.