There are several ways to tackle lighting a fire when facing damp circumstances. Assuming you have a knife, a fire starter and some sort of tinder you are (almost) good to go. Tinder feathering is an effective way to light a fire a keep it going. Start using dry tinder, if you don't have any, use the driest stick you can find; roughly 1" diameter is great. Remove the damp bark by using your knife edge. Start from the bottom of your twig and cut thin slices down and outward into the wood, as to we can get to drier wood. Cut down just enough as so to not fully detach the thin twigs. These protruding thinner slices will burn more readily. This is called a 'feather stick'. Light it and slowly build your fire on top. Adding any dried moss or leaves will help your feather stick burn stronger.