If I put someone in the "ignore list" on TSR would they be able to see my posts?

Yes, I say that the ignore feature is rubbish on this website for several reasons.

For this one though?

I don't think you should be able to post in public but then expect that what other specific users see is forcefully censored, that's kinda dystopian/Black mirrory. Having TSR hide certain users posts/messages from you is one thing but editing another users access to public posts is another IMO Especially if we aren't in a situation where any forum rules are broken and it's a straight user request.

Considering how easy it is to sign up for a new account anyway I think the OP would need to reconsider what identifying info they were posting in public if it was a concern rather than expect to obstruct other users access.