Do CSS class names have any effect on SEO?

Google says nothing about that. The more important things are page structure (correct tags, H1 for headers, SECTION for sections, etc..), and also metatags and urls.

Google will give visibility to sites that loads fast, have good document structure and descriptive urls. You should focus on that.

More info here: http://support.google.com/webmasters/bin/answer.py?hl=en&answer=35291

Absolutely 100% NOT!

It's a good rule to make your CSS classes useful and descriptive for a human to read...but what you're seeing here is probably just this; good naming conventions. If your competitor believes that this will help with SEO then they are mistaken.

Instead, focus on good 'in page' content, descriptive titles and useful meta tags.

Avoid keyword stuffing and spamming; it will get you nowhere fast.

Good luck, Michael.

Answer: Do not use custom CSS class names for SEO benefits.

Many modern search engines will not be looking for keywords within your tags - imagine the miniscule percentage of websites that actually customize their tags to cater for search engines and how much unnecessary work would be required to implement tag-crawling as a result.

Instead, make sure to use standard tag names such as a, ul as much as you can without having to result to CSS tricks. These will make your site much easier to crawl, indexing many more pages as a result. Make sure to use heading tags (h1, h2 ... hn) as much as possible over custom CSS font tags as search engines will make sure to emphasize these results later.