The president's full name in the movie is William Harrison Mitchell; William Henry Harrison was a real president elected in 1840 and the first president to die in office. The vice president in the movie is named Nance; John Nance Garner was Franklin D. Roosevelt's vice president for two terms, and once derided the job as not being "worth a pitcher of warm piss" (bowdlerized at the time and often misquoted as "spit").
The House of Delegates Chamber in the State Capitol building in Richmond, VA doubled for the House Chamber of the U.S. Capitol in Washington, DC. While the movie was being filmed, many members of the Virginia House of Delegates and visitors to the Capitol that day were used as extras.
United States Senator 'Alfonse D'Amato' was set to cameo as himself but then pulled out of the film following issues between the Senate and Warner Brothers over the content in Ice-T's album "Cop Killer". D'Amato later attempted to return to the film after Ice-T parted ways with Warner Brothers but director Ivan Reitman turned him down.
The trivia items below may give away important plot points.
When they are swearing Gary Nance as president, they refer to him as the 45th president of the United States. This means that Mitchell would be the 44th. George W. Bush is the 43rd President of the United States. Barack Obama is the 44th President of the United States.