Can you tour the FBI building?
Tours are available of "The FBI Experience" but requests must be made in advance through the office of a U.S. Representative or Senator. That's similar to what one must do in order to tour the White House. As a non-citizen, you are not prevented from requesting a tour, but you would have to find a congressional representative who would accept your request.
If you want to pursue this, I suggest you read the Top Question (full site, not mobile) about how to tour the White House as a foreigner for some tips on finding such a congressional representative as well as the info about FBI tours on the link below.
I don't believe non-citizens can tour the FBI building unless they have a green card. This is on the page longhorn linked to: "All visitors to The FBI Experience must be U.S. citizens or valid green card holders."
Whoops--ignore my previous post. I missed the part in the FBI info which said tour participants must be U.S. citizens or green card holders. You aren't eligible to request a tour of The FBI Experience. (The same is NOT true for requesting a tour of the White House. Foreign visitors are welcome to apply for those tours.)Edited: 12 October 2017, 12:50
Ohh thanks for link.. boo. Oh well!
That's unfortunate. In the early 1960s, I remember the FBI Tour was a very popular favorite for boys and men. During the Cold War, and after release of the book/ Hollywood film "The FBI Story." Long before Mr. Hoover became discredited for his overstepping of authority.Edited: 14 October 2017, 23:36
The tour was also popular with "girls and women".