What would happen if you had x-ray vision and closed your eyes?

That depends on the strength of the X-ray vision, is it possible to change the intensity?
If the answer is yes then you could probably turn it down to the point where you could ignore any background images and sleep, or even turn it down completely.
If the answer is no, then the intensity might be such that you could see right through everything and not really be able to make sense of your surroundings.

This is a hypothesis, in reality (outside of comic superheroes), the answer is they would be blind. X-rays do not reflect off of surfaces like the visible spectrum we see with and so the eye would only receive from bodies that emitted X-rays, for example the sun, stars and distant galaxies.