<p>I wrote an application that currently runs against a local instance of MySql.  I would like to centralize the DB somewhere on the Net, and share my application.</p>

<p>But, I'm cheap, and don't want to pay for it.  Does anyone know of a free on-line relational DB service that I could connect to via C#?</p>

I've never heard of such a thing. A few years ago, many hosts (Brinkster, etc) offered some minimal database capacity with their free web hosting accounts, but I think you'd find it difficult to find something like that now. Even if you could, most hosts no longer allow you to connect to a database (even on an account you're paying for) outside of a web application running on their server.
My advice would be to cobble together an old computer and use that as a database server that you run out of your house (coupled with no-ip or some similar service, probably). If you're going to need more horsepower/bandwidth than that, you'll probably just have to suck it up and pay for something.