Here is a nice video which talks about seasoned firewood. It's not good to burn wet wood. It will create creosote build-up. Creosote build-up is not good. It can lead to a chimney fire if left unattended. If you have a stainless steel chimney liner in your chimney you still need to clean your wood burning chimney regularly. If you have a corrugated chimney liner the build-up can be more extensive.
Wet wood will burn. It will just burn poorly. It will produce a lot of smoke, and won't produce as much heat as it should. Many people like to turn dampers down at night so that the fire doesn't go out while they sleep. This is really a bad idea if you want to keep creosote build-up down.