I am a real life wildland firefighter and I can tell you that forest fires can definitely burn in the rain. It's quite common.
Large fires can be thousands of degrees in the interior, hot enough to melt steel. Rain can't even touch it - the raindrops evaporate long before they touch the fire. Even a single torching tree is hot enough to evaporate raindrops.
Of course, fire is complex and rain certainly helps to put it out. It depends on numerous factors. In Colorado, the fuels were receptive to rain and once it started raining, we could usually sit back and declare the situation under control. In Alaska, when the rain came down, it was the exact opposite - we would often kick into high gear, because rain was our chance to catch the fire. Those were some miserable times.... slogging through knee-deep bog water as the black spruce torched all around you
Now, as for starting the fires.... that might be a different story, it just depends on the skill level of the person starting it. I've started plenty of campfires in the rain before, but it's certainly harder. Using only prehistoric methods would increase the challenge even more. Still, with enough fuel, you can burn anything...