The Mediterranean loses more water to evaporation than it gains from rivers and rainfall. It's only at sea level because it's connected to the Atlantic at the Strait Of Gibraltar. Dam the Strait, and the Mediterranean goes down.

That isn't just speculation, the Mediterranean actually was a dry sea, when the Strait closed 5.9 million years ago: http://en.wikipedia.org/wiki/Mediterranean_Sea#Geology

A better question is "how quickly would the Mediterranean dry up if you dam Gibraltar?" A thousand years? More? Would you have to dam every river that drains into the Mediterranean?

If the dam's levees were closed, the water in the mediteranian would evaporate like it always does, but no new water would flow in from the Atlantic.

This has happened before. From 5.9 to 5.33 million years ago, the strait was closed and the entire sea evaporated. It will happen again, when the northward moving African plate collides with Europe.