There are all sorts of threads going on here — about tourism and the vision of what a place stands for and fear of change. But those strike me as a separate matter from the anti-discrimination law in question. Unless they are really looking to draw Christian tourists specifically as "The Town Where Its Okay to Discriminate Against Gay People."
In Arkansas, Gay Rights Ordinance Highlights Clash Between Two Faces of Tourism – NYTimes.com
A measure supporting gay rights in Eureka Springs is on a May 12 ballot, and opponents say the effort to promote the town as gay-friendly has hurt a nearby Christian destination.
Tourism built on hate was never a sustainable business model in the first place.