When Chuck Barris is pitching The Gong Show in 1977, someone is drinking from a mug with an NBC logo that shows a peacock over a stylized "N". This logo did not exist until the fall of 1979, and was replaced with the current logo in 1986.
When The Newlywed Game was picked up and everyone was celebrating, there was an episode of the show playing on the TV in the other room. That particular episode was actually from more than 10 years later.
In one scene, with Chuck's secretary and Chuck Barris in his office, the scene is labeled as taking place in LA, 1967. On a close up of the secretary, over her left shoulder, is the sheet music to Frank Mill's "Music Box Dancer", which was not written until 1974, and didn't become a hit single until 1978.
Twice in the film Chuck Barris lights a cigarette backwards (i.e. he lights the filter with the tip in his mouth.) The first time is in Chuck's office with him and his secretary (labeled as taking place in LA, 1967), and the second time is in a car with Chuck, Byrd, and Simon Oliver in Helsinki 1967. Also, in each of these scenes Chuck is smoking a different brand of cigarettes, though this may very well have been on purpose.
When Chuck Barris and Byrd are in Mexico City (according to a caption), we can see a Saguaro cactus in the background. This type of cactus can only be found in the Sonoran Desert in the states of Arizona, USA and Sonora, Mexico and is rarely found growing over 4,000 ft. Mexico City is not in the Sonoran Desert and is at 7,500 ft elevation.
When the new CIA agents are target practicing, they are a shooting range that has a clear flat terrain behind it as far as the eye can see. That would never be the case. There is always a hill or some manmade elevation behind the targets to "catch" the bullets. Otherwise there'd be strays all over the place.
When Chuck Barris shoots the man in Helsinki ("sorry 'bout your teeth") he is using an automatic pistol, but no cartridges eject. Only revolvers keep the cartridges inside the gun after the bullet is fired.