Yes, this is the classic gentrification problem. Now you've turned a restaurant that locals can visit into a purely tourist attract that only the upper class can enjoy. This solves the problem of the restaurant owners having to close down, but for all the locals the restaurant might as well be gone.

P.S. And then when the next "top N burger joints in America" list comes out that everyone goes crazy for and all the burger-tourists start going somewhere else now you've lost your local customer base and you might go out of business regardless. Raising prices works well if you're a fine dining establishment in a dense metro area and that's the kind of business you expect to run. It doesn't work so well for burger joints or BBQ shacks elevated to a bizarre level of temporary fame.