The reason most t-shirts do not wrinkly and napkins do is because t-shirts are knit and napkins are woven. Whether or not wovencotton fabrics wrinkle depends on the way the fiber is woven, the thickness of the yarn, the finish on the yarn or the fabric. You can usually tell if a cotton napkin is going to wrinkle in the dryer is to scrunch it up in your hand before you buy it. If it forms deep wrinkles it's going to wrinkle in the dryer. Same for woven clothes.
Taking them out of the dryer while still a bit damp and smoothing them out will work for some napkins. They have to be a bit damp for this to work. How damp depends on the fabric. I dry some of my blouses in the dryer until they are just barely damp and then hang them up, smoothing the placket, collar and seams.
My grandmother never ironed. She also did not have a dryer. She hung all of her clothes on a hangar and "ironed" them with her hands. They were always presentable. She did have one of the first front loading machines and used a small amount of detergent so that the clothes were completely free of detergent residue.
I also like the idea of smoothing our the damp napkin and placing it over a plastic hangar to dry. You can then fluff them up by putting them in during the last minute or so of a dryer load if the load is not overly or bone dry.