When my 8889-T form becomes available on 01/23/2020, will it affect what is currently showing for my estimated federal and/or state refunds?

It may.  I believe you are asking about a 1099-SA that you will receive in order  to complete the Form 8889 and am responding to the question accordingly,

 

You need to file 8889 to:

• Report health savings account (HSA) contributions (including those made on your behalf and employer contributions).
• Figure your HSA deduction.
• Report distributions from HSAs.
• Figure amounts you must include in income and additional tax you may owe if you fail to be an eligible individual.

 I would hold hold off on filing your return until you get your 1099-SA.  Waiting could potentially save you a considerable amount of time filing an amended return.    

 

Form 8889 is expected to be available in TurboTax on 1/23/20.  To verify if the form is available follow this link:  Form Availability  

 

Should you have additional questions in regards to your question, please respond to this thread.

2019 TurboTax appears to be doing all of the correct calculations on Form 8889 and no changes are anticipated, so what you are presently seeing on this Form 8889 should remain unchanged when the form is finalized.  Finalizing the form with the IRS seems to be just a formality that remains to be completed.