Does the $39 processing fee apply if I pay with either my credit card OR refund, or just if I pay with my refund?

If you pay your TurboTax fees upfront using a debit or credit card you will avoid the EXTRA "refund processing" fee that is charged by the third party bank if you want your fees to be paid from your federal refund.  Pay upfront, avoid the extra $39 fee, and have your refund sent directly to your bank account without involving the third party bank.

There is an Extra $40 Refund Processing Service charge (45 in California) to have the fees deducted.   Which you can avoid by paying first with a credit card.

How to review your fees

https://ttlc.intuit.com/community/charges-and-fees/help/how-do-i-review-my-fees-in-turbotax-online/00/26353

What is the Refund Processing Service

https://ttlc.intuit.com/questions/1900596-what-is-the-refund-processing-service

Remove Pay with Refund fee

https://ttlc.intuit.com/community/downgrading/help/how-do-i-remove-pay-with-my-refund-fees-in-turbotax-online-and-change-my-payment-method/00/25880