1. Look at Yelp's terms of use to see who owns the rights to their posts. It's either the poster, Yelp or some combination of both with licenses.
2. Would short excerpts fall under fair use? I would tend to think so but this is a large gray area so I won't say for sure yes or no. The shorter the better. You could probably say, e.g. "A user on the popular Yelp website said that "the Japanese food was so yummy I ate there 50 times last week!" A full paragraph or two might be pushing things too far. Also consider the number of quotes you'll use. If it's 8 in a good sized book, that might not be too much. But if every other review has Yelp quotes, expect to hear from them. With my accreditation, I mention this from the practical experience dealing with the business issues (high importance) in having policed copyrighted property. If we received reasonable accreditation then the minimal possible infringement wasn't an issue. Can't speak for everyone.
3. Do you need permission from Yelp? See #2. The less you use the more likely you may have a fair use claim and less likely Yelp will think any lawsuit is worthwhile. If you want to obtain their rights to use the quotes, be prepared to provide them with some compensation in some fashion and I'm guessing it may be more than you might care to provide. You can always try but then you'd be alerting them to something that they might want to police more strongly.
Just some thoughts. If you're thinking of using many of Yelp's reviews... then you really want to be careful. Good luck.