I am looking for a film from the 70's or 80's where the president of the United States of America is kidnapped (possibly kept underground) by a group who then produce fake videos (using some type of photo-realistic CGI) of him to run the country.
You may be thinking of the 1973 Woody Allen film Sleeper (IMDB). Woody Allen plays Miles Monroe, who was cryogenically frozen in 1973 and revived 200 years later. America is run by an authoritarian government, lead by a dictator called "the leader", and Miles falls in with some rebels trying to take down the government.
We never see the leader speak or make any public appearances. He is seen a few times in photographs and generic footage with a narrator. I understood the depiction of the "generic footage" to be in-universe stock footage of the leader, rather than CGI.
During the final part of the film,
we learn that the leader was killed almost a year previously by a rebel bomb. Scientists managed to save the leader's nose, and plans to clone a new leader from the nose. Monroe manages to steal the nose out of the operating room, is chased by the authorities, and finally assassinates the leader by throwing the nose under a futuristic-looking road roller.