Your question, though a little old, is still relevant, and what started out a question about APIs turned into one about duplicate content. If you asking whether using APIs themselves will hurt your SEO, by themselves they wont. However, depending on the implementation, it could hurt SEO, but it probably still wouldn't be the fault of the API ;)
As for the duplicate content, theres a lot of fluff on the interwebs on the topic. It's not so much that it's bad for all SEO, but a bad thing for search engines in the quest to serve the most relevant content. The problem is that it's everywhere and there's no stopping it (search engines openly recognize this), except where it's most important, which is on your own site. Let me put it like this, A higher authority site can take content from low authority site and can usually outrank it every time. Who would you say got the penalty there? Conversely, If your site has several pages that are the same or deemed close enough, you will indeed get a penalty.
Think about it, some of that content mentioned from yelp, is likely dupe content from the business site itself, as surely other content they pull in from other apis is too. Believe it or not, some sites can and do subsist off of largely all content from other sites aka duplicate content (alltop,dzone,reddit).
Related to the question, I say use the APIs, but just make sure you layer in other content, hopefully unique/relevant keyword rich. Good Luck!