Professor Michael Luca of Harvard Business School and Professor Giorgos Zervas of Boston University have released an independent study debunking the allegations that Yelp provides preferential treatment to advertisers. (For what it’s worth, we have always responded to this conspiracy theory the same way: it is not and has never been true.)
In the professors’ words, “Yelp's current implementation of the filtering algorithm does not treat advertisers' reviews in a manner different to non-advertisers' reviews.”
You can read the relevant section of the study here (or read the full study here):
The TL;DR version? Tests were run to determine if a business’ decision to advertise on Yelp had any influence on which reviews were recommended. Professors Luca and Zervas ran tests on a dataset of 316,415 reviews for 3,625 restaurants to determine if a business’ decision to advertise on Yelp had any influence on which reviews were recommended. No correlation between the reviews displayed on Yelp business listings and advertising was found, or as the authors put it, “neither 1- nor 5-star reviews were significantly more or less likely to be filtered for businesses that were advertising on Yelp."