Using MyISAM tables is not required for basic functionality, but your results will be better with MyISAM. Here's why:
YARPP uses "fulltext" indices in order to do comparison of content and title keywords in a performant way. Unfortunately fulltext indices are only available with MyISAM tables. (InnoDB fulltext support is coming in the future, the MySQL project tells us.)
The short version of how MyISAM and InnoDB tables are different is that, if you are on a very high volume site where many transactions are occurring at a time, MyISAM may have to wait for other transactions to complete before others move forward, while InnoDB can do things in parallel better. This is a *HUGE* simplification. For most sites (if you're not a dba administrating tweaking MySQL server's settings) switching your posts table to MyISAM shoud not be an issue.
If you want to use YARPP with your InnoDB posts table, I suggest lowering your "match threshold" and making sure the "body" and "content" are not being considered in the "relatedness" algorithm. Go to YARPP settings, "screen options" tab, show the "relatedness" settings, and lower the match threshold and save. YARPP will just rely on tags an categories in common.
I hope that helps.