what happens to my published project once my Adobe Captivate subscription is over?

There shouldn't be any issue with your published project when an actual subscription expires.

Projects published with the TRIAL version will expire (not run anymore).

And, of course, you won't be able to open your source file until you reactivate the subscription, which can fortunately do monthly if you just need to make a quick update.

However, what happens if you don't touch the source for a couple years and, the next time you need to, Captivate has advanced enough vesions that it won't support/upgrade your source file version anymore?

That could be an issue.

But your published pieces shouldn't be affected at all (at least, as I understand it).