After paying the fee for e-filing the state return, I learned that I cannot e-file because my federal return has to be sent regular mail. How do I get a refund?

You can go here TurboTax CD/Download Refund Request Form to request a refund if still within 60 days of purchase. 

To contact TurboTax customer service/support use their contact website during business hours.  If the problem concerns any type of refund for fees, use the key words billing issues and do Not use the word Refund or you will get a phone number for tax refunds.

Submit your question/problem during business hours and then use the option for Call (a phone number will be provided for Call) to speak directly with a TurboTax agent –

https://support.turbotax.intuit.com/contact/