In this fascinating article, the emissions from growing trees for paper, cotton and linen; the emissions from manufacturing these materials, as well as the emissions from washing and drying the cotton & linen napkins are taken into consideration. The conclusion?
Cotton is not a very earth-friendly plant:
"each 28 gram cotton napkin causes over one kilogram of greenhouse gas emissions and uses 150 liters of water! By comparison, the paper napkin causes a mere 10 grams of greenhouse gas emissions and uses 0.3 liters of water use while the linen napkin causes 112 grams of greenhouse gas emissions and uses 22 liters of water." So ixnay on the ottoncay apkinsnay.
What surprised me is that paper napkins are the preferred napkins IF they are going to be used only once. Restaurants that have fabric napkins, of course, wash them after every use, which is not water-saving at all.
So the ultimate napkin is a linen one that you use multiple times (suggested: a week) before washing, and don't forget to bring it to restaurants.