The IRS and your local state have separate refunding procedures and schedules, which just means your refunds are processed separately. So paying your TurboTax fees from your federal refund doesn't affect your state refund at all.
And if you selected Refund Processing Service to pay your TurboTax fees out of your federal return, Refund Processing won't slow down or speed up your federal tax refund. That is just an easy way to pay your TurboTax fees.