Great question Miriam. I just wish Yelp was asking me.
First thing Yelp should change is their perception of businesses. Their attitude sucks, and businesses feel it. Is it any wonder they're known as the online mafia? This would require something as basic as Yelp changing how they word things when talking about consumers and businesses.
Second thing would be for Yelp to understand that not everyone is a Yelper. As such, it's okay for businesses to legitimately ask for reviews. "Check us out on Yelp" just doesn't cut it. If Yelp was in more favor with businesses, and businesses were permitted to ask for feedback on their profile, Yelp could turn businesses into an audience of promoters. In turn, this would increase Yelp's traffic and volume of reviews. Even the number of Yelpers would increase.
Finally, Yelp should invest in a dispute process. It would likely produce a higher ROI compared to the home services function, for the same reasons I mentioned above. I can't tell you how many businesses I've worked with who obtain reviews from people who were never their customers or patients. For example, a plumber knows who his/her customers are. If the plumber sees a name attached to a Yelp review that isn't a customer, then the plumber should be able to dispute that review, even for a small fee. The customer could then upload a receipt from the plumber if they had one to counteract the dispute. It's a scaled back version of the "No Receipt No Review" point of view (thanks TomEats), requiring a receipt only in the case of a dispute.
I believe the outcome of all of the above would produce more Yelpers, happier businesses that would be more willing to advertise, and more authentic reviews. Until then, Yelp is leaving a host of revenue on the sidelines.