<p>What is the most efficient way to determine how many comments a particular blog post has? We want to store the data for a new web app. We have a list of permalink URl's as well as the RSS feeds. </p>

If the blog is controlled by you, a "Select count(commentid) FROM comments WHERE postID = 2" will possibly the best thing. If you only have the URL but still it's your blog/db, you need to create a subquery "WHERE postID = (SELECT whatever FROM posts WHERE permalink = url)" or whatever your way to join the comments to the posts from a URL.
If it's a remote blog, you have the problem that each blog has different HTML. Essentially, you're going to need to build a parser that parses the HTML and looks for repeating elements like "div class=comment". But that will be mostly a manual labour for each different blogs.
Some blogs may have better ways like a comment count somewhere in the HTML or some interface, but i'm not aware of any standardized way.
EDIT: If you got a Comment-RSS feed, you
may
have luck using a mechanism that counts XML nodes, like XPath's
Count
.