Depends what it is
Most will at the very least degrease the skin and make it hard and flaky. Alcohols will do this. Contact dermatitis can result from that or from a reaction with more aggressive solvents.

Some solvents are quite benign to humans, like ethyl acetate. That's why it's replaced acetone in nail varnish remover.

But you also get quite a few organic solvents which are pretty nasty. The chlorinated ones, for example dichloromethane (found in paint stripper), aren't good for you at all. They can also pass through the skin quite readily and into your system, where they can cause cancer and other maladies.

And let's not get started on benzene. Even though it's got a really nice smell.
( K2k6 has a proper job these days, Wed 20 May 2009, 12:17, Reply)