Resinous wood is king in wet-weather. 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, hickory, etc…. 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 during our rainy season (yes, we get rain here and lots of it!). 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. This was a tip I learned while attending one of Mors Kochanski's fieldcourses in Canada during the late 90s and it has served me well over the years. The interior wood on such a tree will be dry and can be shaved into fine pieces (tinder) to ignite your fire and then the rest of the log can be burned.