The park bench that Tom Hanks sits on for much of the movie was located in Savannah, GA, at Chippewa Square. The fiberglass bench he sat on has since been removed and placed into a museum to avoid being destroyed by the weather. The church where the feather first falls is about 100 yards just down the street from his bench.
In the original script, every member of Forrest's platoon in Vietnam was supposed to be mildly retarded like him. This was an experimental practice of the Army in the Vietnam War; the Army tried allowing soldiers with lower IQs to enlist, but put them all in platoons together. Because this led to so many discipline problems, the Army is not particularly proud of it and threatened not to work with the filmmakers at all if they left the platoon like that. As a result, only Forrest and Bubba are noticeably below average intelligence.
Dave Chappelle was originally offered the role of Bubba in Forrest Gump, which was in a pre-production stage. He was concerned about what seemed to be a racially demeaning character so he turned down the part. He has since admitted to regretting the decision.
When Forrest is meeting President Kennedy, the music playing in the background is from the musical "Camelot." The history of this connection is as follows: In a 'Life' magazine interview (a week after the President's assassination) Jacqueline Kennedy (his widow) compared the Kennedy years (with children Caroline and John Jr.) in the White House to King Arthur's kingdom "Camelot", explaining how fond the President was of listening to the musical Camelot's title song.
Tom Hanks' scene where he calls the hotel management from his room to report about the "power outage" across the street in the Watergate was actually filmed on scene within the Howard Johnson's hotel which contained the look out room for the burglars and was the place from which the police were called. The hotel is now a dorm for George Washington University.