How do I delete a rejected state tax return? I filed in my nonresident state where I work by mistake and there is no way to delete the incorrect one that was rejected…

You can delete your state return by following the steps in the below TurboTax help article:
How do I delete my state return in TurboTax Online?

If you read my question, you will see that I asked about a processed and rejected state tax return (already paid for, sent to IRS, etc). Can I simply ignore the rejected state return or do I need to contact someone to have it removed altogether…? 

After you have registered or paid for state software and filed it, we can't delete it entirely from the software. However, if you are not required to file in that state, you can just leave it rejected and it will not cause a problem.