Resinous wood, such as pine, spruce, and fir, piled up on a fire in wet weather will help to sustain it during a downpour. Look for wood that is saturated with gooey sap. This stuff is impervious to moisture and will ignite even when wet. Avoid hardwoods like oaks, maples, birch, and hickory. The latter are great for providing long-lasting coals for campfire cookery but won't burn furiously when wet.
In the forests near where I live in Arizona, we rely on dead Ponderosa Pine trees. It has characteristic yellow streaks that indicate the presence of resin in the wood. In fact, there is probably more resin than actual wood as evidenced by the absence of coals in the firepit hours after burning. We have even gathered limbs sitting in puddles for days (yes, it rains here!) and ignited it. In the Great Lakes, where I grew up, we would always use spruce and pine trees and even gather the balls of sticky sap to create mini-torches. Resinous wood is found the world over, so look to this when the skies are grey and the night is stormy.
One final tip for a wet-weather environment if resinous wood is in short supply: Gather sections from a dead-standing tree over 4" in diameter and split it down the middle with your knife or ax. The interior wood on such a tree will be dry and can be shaved into fine pieces (tinder) to ignite your fire, and the rest of the log burned.
You can see for yourself how to build a survival fire.