Summarize:

If it's raining, or has recently rained and you don't have access to dry firewood, it's still possible to keep your fire burning. It just takes more effort and patience. Focus on building up a small flame in one area of your fire pit. The larger the wet area and material, the harder it will be to create enough heat for a sustainable fire. Use extra tinder and kindling. Don't try to light a whole log just yet. Work on building a flame with paper and sticks. Birch trees have good bark for quick burning even when it's raining as the bark has natural oils which repel moisture. If you can, place a tarp or something over your fire to block any rain. Just make sure it's enough to prevent scorching or catching fire. Use a dry towel or even dry clothes to wrap all of your wood and burning materials in. Damp the wood down and try your best to suck up or get rid of as much moisture as possible. If you suspect you may encounter rain, grab a canister before you go out and fill it with dry twigs, pine cones and needles. An oatmeal tin works great for storing small kindling and will keep it dry. When burning a fire outdoors, it's always a good idea to have extra logs which you keep wrapped up just in case of rain. A grouping of smaller logs and kindling will be easier to light individually than one large log. In addition, you can try to light anything that will burn to get your fire started, or keep it going. Waterproof matches, a lighter, or flint and steel are your best best for sparking a flame. Foods that have a lot of carbohydrates can also be used as fuel in a pinch. Foods like chocolate and marshmallows work well. If you have a hatchet or other way to split logs, use it. Split logs down the middle to expose the dry area. Stand the logs up and face the dry bark towards the flame.
Start small. Wrap your wood in a towel before trying to burn. Use smaller logs, sticks, and anything else at your disposal.