Though not an intentional test, we found out what happens when a movie gets a wide release but does not advertise. In 2008, the movie 'Delgo' was released on over 2000 screens with nearly no advertising. Because the production company could not find a distributor, they spent their ad money on renting the screens for a week, with the hope that some people would randomly see it and word of mouth would spread, leading to the theaters wanting to keep it for additional weeks. It became the lowest earning wide release movie up to that point in time. Each screening averaged two people per screening. More people saw Conan O'Brien making fun of the movie in his monologue than actually saw it in the theater.
The reviews for the movie were poor but it had lots of household names as its voice talent, including Anne Bancroft in her final film. Good or bad, advertising likely would have lead to more people seeing it than two per screening. Of course there's no way to know for sure how many more would have been enticed by the ads but we do know that going with zero advertising resulted in a huge disaster.