Wood combs should not be used on wet hair - the wood will warp and crack.  Horn combs can be used on wet hair, but should be dried after using.  They can get yucky if they stay wet for extended periods of time.

Well, for me it varies. If I'm in a hurry I comb it when it's mostly dry, but often I wait until it has dried completely out of pure laziness.   Combing my hair wet is just gah, it tangles and behaves really badly.