We have bought a few different kinds of cloth napkins for our dinner table and they seem to always come out of the dryer covered in tiny fine hairs. 
How do I get rid of these? I wash them with our kitchen towels and dry them on high with a dryer sheet.
Welcome to electrostatics:
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If you use a dryer, clothes become dry; yet the typical dryer basically is a rotating drum containing the clothes where clothes rub against each other. This creates both fine particles and «hairs» swirling around, and separates electric charges. Some of the particles end in the filter of the dryer, others remain with the clothes which -- because they are not metallic, and not good electric conductors -- do not discharge well with the metallic drum of the dryer. And this renders clothes like magnets for the small hairs and fibers (or, lint); synthetic clothes tend to be more prone to this, than cotton.
This where -- if you intend to use an electric dryer -- fabric softeners come into play, or the dryer sheet you mention to reduce the effect.
If you do not need the table clothes this quickly and have the space outside, or in a well ventilated space (e.g., below the roof, laundry room, basement), consider drying the clothes on a clothes line or rack prior to iron them / folding for the closet / drawer.
Fabric and method of washing usually the reason why as certain fabrics are prone to pilling especially synthetics.
Washing with like fabrics on gentle cycle and allowing to air dry might help. Check the fabric instructions. Depending on what material(s) they are made of heat drying with heat could be your problem.
Also, in terms of washing, what type of washing machine and how full your loads and what you wash them with can all play a big part too. Top loading washing machine tend yo be harder on fabrics then front loading machines.
As to removing the pilling a razor works great to remove the little pill fluffs.
Perhaps just let them air dry.
