tl;dr Only push comments with # on them from Salesforce to JIRA. Or expose an API that would allow customers to customize this.
Separately, we’ve been looking at ways to get comments passed from SF to Jira, and our fellows here had an interesting solution – I set up an account/jira api comment call for them to use, and they’re having a hashtag in salesforce send a given comment to jira. There are various issues with this which we’re addressing, but as we were working on it, I got to thinking that SR is likely doing something similar when it’s syncing comments from a remote source? If that’s true, do you have an API call (akin to the Endpoints links) that writes the data contained as a comment on all tickets associated with a Case ID? We could adapt the hashtag posting mechanism to use the SR URL. It’d be a way to selectively pass some comments from SF to Jira without a button to push it. Would it be best to pose this to your support?