Following Chaplin's 1921 trip to England, there's a scene with Charlie and his pal Doug Fairbanks (Kevin Kline) who refers to visit from "the FBI". In the 1920s what later became known as the FBI was simply called the Bureau of Investigation, an office of the Justice Department. The Bureau of Investigation was renamed "Federal Bureau of Investigation" in 1935.
During Chaplin's 1921 trip to Great Britain, he rides a Southern Railway train, which pulls into St. Pancras Station. Southern Railway didn't exist until 1923. St. Pancras Station was owned by the Midland Railway, which became the London, Midland and Scottish Railway in 1923.
Towards the end of the movie there is a shot of the New York waterfront with the French ocean liner SS Normandie in the foreground, the subtitle showed the year as 1952. The Normandie was seized by the US government in 1942 and as she was being converted to a troop ship, caught fire and capsized at her dock. She was sold for scrap four years later.
When Charlie's brother and others arrive to whisk Charlie and his unedited film away to Utah, the car screeches to a halt. However, the car is on a dirt track and therefore no screeching noise would actually be created.
When Douglas Fairbanks swings down on the rope to meet Charlie, his foot is inside a loop at the end of the rope when he starts the swing in the wide shot, but when he lands in the tight shot, his foot is nowhere near the loop. Also the momentum created by the swing would have carried him much further than it actually did, i.e he wouldn't have stopped where Charlie was standing.
During the scene where Chaplin demonstrates the power of silent films as Vaslav Nijinsky, he first gives him a high voice before correcting himself. Chaplin would've known very well what Nijinsky sounded like as he had met him numerous times in the 1920s and the two were good friends. (Source: The documentary "Chaplin Today" from the DVD release of City Lights)
When Charlie and Mildred Harris are in the bedroom, Charlie wants her to put lipstick on. She uses a metal tube twist up lipstick. The scene following this bedroom scene is dated as 1918. Twist up metal lipstick tubes were not invented until 1923 by James Bruce Mason Jr. in Nashville, Tennessee.
When Charlie is shown at the 1972 Oscars near the end of the movie, he is showed being brought to the podium in the wheelchair and then standing at the podium as the movie clips played. In the actual awards ceremony, Charlie walked out to the podium under his own power after the clips ended.
When Charlie and his brother visit Fred Karno for Charlie's "interview", the opening shot shows a river with still water and a paddle steamer moves from left to right. The boat is moving a good deal faster than the paddle wheel is moving suggesting another power source.