Is it true that if we pay for our TurboTax fees out of our tax return then we are ineligible for an early refund?

If you are referring to the advance loan, that ended on February 15.

Not true.  How you pay TurboTax does not affect the processing your tax return with the IRS.  IRS processes the return regardless of how you pay your fees to TurboTax.  TurboTax and the IRS are two separate entities.