In one scene labeled as taking place in Los Angeles, 1967, on a close up of Chuck's secretary, over her left shoulder, is the sheet music to Frank Mill's "Music Box Dancer", not written until 1974, and not a hit single until 1978.
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.
Twice in the film Chuck 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 Barris and Byrd are in Mexico City (according to a caption), a Saguaro cactus in 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 at elevations over 4,000 feet. Mexico City is hundreds of miles southeast of there and at a 7,500 foot 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.
The same extras are used for different scenes. When Chuck is in the cinema you can see the same man as later in the audience with one of Chuck's quiz shows. This is likely deliberate, given the odd humor of the movie.