Is there a famous museum about conspiracies?

It is specific to one type of conspiracy, but the UFO Museum in Roswell, New Mexico is interesting.

I will warn you though, the museum isn't all that fancy. Much of it looks like a high school science fair. They clearly aren't a big budget operation, but it is worth the cost of admission at least.

Also about one particular event, the JFK Sixth Floor Museum in Dallas does a very good job covering the conspiracies about JFK's assassination.

In Washington DC, there is the Crime Museum. It is near the Spy Museum.

There is also an exhibit at the Newseum that might be up your alley.