Paper napkins have been around since the invention of paper, so they definitely came first. The Chinese used them when drinking tea. Still, it's an interesting question about how it related to the American fast food industry that I'd like to see answered

American usage of paper napkins preceded fast food restaurants by about ~30 years. Paper napkins are believed to have been introduced to the USA in 1885 http://restaurant-ingthroughhistory.com/tag/napkins/ This seems to be corroborated by this patent https://www.google.com/patents/USD162097

http://www.wisegeek.org/what-was-the-first-fast-food-restaurant.htm

The first fast food restaurants likely modeled themselves from carnival vendors, who would not have the capacity to wash linen napkins. Perhaps this was popularized during the 1893 Columbian Expo https://books.google.com/books?id=pz41AAAAMAAJ&pg=RA1-PA102&dq=1893+Columbian+Exposition+paper+napkin&hl=en&sa=X&ved=0CCwQ6AEwAGoVChMIxr6E8JmXxwIViz8-Ch2Wfgzs#v=onepage&q=1893%20Columbian%20Exposition%20paper%20napkin&f=false

Of course, it is said that George Ferris drew his design of the 1893 Compian Expo Ferris Wheel on a paper napkin, so they were probably commonly used in banquets and other places where many people consumed food before then. https://rockypointpark.wordpress.com/tag/columbian-exposition/

Not sure if this is uesful or interesting, but I didn't want my trip down the rabbit hole to be for nothing.

Interesting question, what makes you think there is a relation?