Wikipedia's list of oldest companies says that Stiftskeller St. Peter, an inn within the monastery walls of St. Peter's Archabbey, Salzburg, Austria, is the oldest restaurant (dating to at least 803), but hotel Nishiyama Onsen Keiunkan has presumably been selling food since its founding in 705.

It's difficult to define exactly the boundaries between tavern, tea house, and restaurant, but I'd be curious what the oldest business serving prepared meals but not offering accommodation.

[0] https://en.wikipedia.org/wiki/List_of_oldest_companies