I had this question fired at me today on a telephone conversation with a prospect.
The answer is the Cassini search is not affected by HTML templates. You only have to see the recent blog on the World’s Top eBay Sellers to see this isn’t of concern. However, if the template isn’t mobile responsive then this could affect conversion rates. This in terms of it showing up at all in the search results on your device rather than having any HTML code in the template listing.
The blog I am currently working on is how you can crack the Cassini search to get your sales back on track.