What kinds of alcoholic drinks are safe for people with celiac disease? Potato vodka? Gin? Tequila? Wine? Rum? Rice wine?

Many options... All the things you listed are viable options. Regular wines produced from grapes are allowed. I have seen gluten free beer, but i don't know how palatable it is. See this website for an extensive list.http://www.celiac.com/articles/222/1/gluten-free-alcoholic-beverages/page1.html.
Celiac drinks. Bourbon, champaign, brandy, mead, ouzo and many more.