It's a bit of a catch 22. The only way to get wet wood to burn is to dry it out first, which means you need a fire to begin with.
If your at a shelter with a fire ring, there is always a lot of little sticks around the front of the shelter. Use these to get the fire started. Damp wood is only damp on the outside. Shaving a bit of the outside off or "feathering" it with your knife will expose the dry part of the stick. (That's one reason why we carry a sturdy knife)
Try to find tree branches on the ground which aren't actually on the ground. If you can make the branch snap, it's dry. It might be wet or damp on the outside, but is otherwise dry.
Unless you have a way to split a log, trying to burn anything bigger then a few inches around is futile. If you can't snap it across your knee, it's too big (or too wet/green) to burn.
In the North East here, White Birch bark makes great fire starter. There never seems to be a White Birch tree near campsites, so I collect pieces during the day along the trail.
Funny story:
I did a hike on the Vermont section of the AT a few years ago in the fall. I was paced with a couple of woman who liked to heat up some Spam for dinner, so I got into the habit of making a fire for them each night. One night close to Hanover, I had a small fire going in a light rain.
A collage ordination group shows up a little before dark and set up a big tarp near-by. A little while latter two young girls (who look like they just stepped out of the mall) come over to the shelter where I'm nursing the fire and ask me "Sir, where can we find dry firewood". Being the wise guy that I can be, I answer "What? You didn't bring any with you?" The expression on their faces was priceless.
I then tried to explain the wood was only damp on the surface and to find down branches, etc., but they never did get a fire going that night.