TurboTax does not pay any type of taxes owed, federal or state.
You pay the taxes owed.  If you selected direct debit from a bank account for the taxes owed, then the bank routing and account numbers are sent with the tax return that was e-filed or mailed.  Only the IRS or State can debit your account.

I paid Turbo Tax via direct debit, Federal Taxes due were paid via efile and TurboTax; however, my payment for efiling for State taxes due were rendered to Turbotax, via direct debit.  Federal payment of taxes during efile were accepted via direct debit, Turbo Tax also charged me the standard $19.99 state efile fee and my state taxes due via efile process. So my question still stands, as I paid my state taxes directly, how do I get reimbursed for the charge to my account for the additional fee Turbo Tax stated they would cover, via charges to my account, for state taxes rendered by me. Perhaps I was not clear enough in the limited characters available for the primary question.

If the direct debit from your account for the state taxes paid with the e-filed return were never debited from your account and you paid the state directly, then I do not see the problem.  Or were the state taxes actually paid twice?  If that is the case then the state owes you for the additional payment and should refund to you the additional payment that you made.

Some states do not let you select direct debit on your tax filing.  Rather, they require you to mail a check or pay on their website.  TT would have given you the options applicable to your state.