"Casablanca" was the first film shown at the new Warner Bros. Theater at the National Museum of American History on Friday night.
There will also be screenings of "The Maltese Falcon", "The Treasure of Sierra Madre" and "The Big Sleep."
Bogart's suit worn in "Casablanca" is also on display, on loan from Warner Bros.
Stephen Bogart, 63, said he didn't really know his famous father because he died when his son was just 8 years old.
Still, he said his father would have been shocked to find himself featured at the Smithsonian.
"He was not full of himself. He was just an actor and hung around with writers and producers and film people," Stephen Bogart said. "Obviously, he would have been proud and would have been humbled, but he would have been shocked."
The American Film Institute has ranked Humphrey Bogart as the greatest male legend from the screen.
To mark the 70th anniversary of the 1942 iconic film, a re-mastered version of "Casablanca" will be released on Blu-ray in March, Bogart said.
The Associated Press contributed to this report.