As far as I'm aware the US doesn't actually ban (criminalize) the mere act of going to any country. However, spending money in some countries may violate trade sanctions and is thus illegal, Cuba being the most famous example.

On the destination country side, North Korea does not generally permit US citizens to visit as tourists, although there are exceptions (eg. the Arirang games in 2005). I can't think of any other country with a blanket anti-US policy -- even fellow Axis of Evil member Iran is happy to admit citizens of the Great Satan.

As for the dual-passport thing, as long as you don't screw up and try to use the wrong passport in the wrong place, one country will never know where you use the other country's passport.