Are there any laws that prohibit the use of headphones while operating a vehicle?
There's the question whether it is illegal per se, or illegal for other reasons. It is always illegal to drive without proper attention to road and traffic. If you are involved in an accident, or if you are stopped by police because they didn't like your driving for some reason, wearing headphones and listening to music or audiobooks may be taken as evidence that you were driving without proper attention to road and traffic.
And as Dawn said, driving with headphones is illegal per se in some places. And I'll add that driving with headphones means putting your life at risk, which is why I won't do it, and it is putting other people's life at risk, which is why you and I both shouldn't do it.
Although the answer has already been accepted, I found a great reference on this that may benefit others that end up at this question for their own state. Each state is listed with a brief sentence describing the law below.