Cotton wicks moisture just fine. The problem is that it doesn't get rid of it, at least not in
our humidity. Maybe in Arizona. I think your locale's gonna play a big part there.
When my cotton shirts get soaked, it feels like I have a big heavy wet towel on me that doesn't breathe; and tucked in my saddle no less. If it was a cool towel, well that'd be great, but it feels more like a steam tent. The second I take it off and change into a dry shirt I'm happy again. Feels like the damn thing weighs about 5 lbs too.
Straight poly or nylon is basically wearing plastic like treezybreez said. But I have to agree, a wet cotton shirt would cause you just as much pain near a fire because steam burns suck. I used a damp towel once to grab a hot piece of metal. Mistake! Grabbed it with a dry towel, no problem. But you might have bigger problems than your shirt in a fire
I can see treezy's point though. I was loading wood into a fire recently with my skidsteer, dead pine - with a bunch of brown needles, just after the recent ice storms. Well I got on the wrong side of the fire. A big breeze lit those needles up hot and blew towards me. Glad I had my hardhat on. Singed all the hairs on my arms and melted some holes in my shirt. Scary as hell! I backed out of there fast. I wont make that mistake again. But the spots where the shirt melted stuck to my skin like napalm. You wouldn't want that to happen with the whole shirt, I promise.
My favorite shirts so far have been triblends - cotton, poly, rayon. Feels more like cotton, has a little stretch, and dries more like the synthetics.
I've heard some good things about those fishing shirts with the vents near the shoulder blades but I haven't tried them.