No.

Generally, The only thing that will change in refinancing usually is the amount of mortgage interest you will be able to claim on the rental.

Expenses you pay to obtain a mortgage on your rental property cannot be deducted as interest. These expenses, which include mortgage commissions,abstract fees, and recording fees, are capital expenses. You may amortize them over the life of the mortgage .

You will have a screen that ask if you refinanced [mark YES]and then on next several screens  the information needed for Turbotax to calculate correctly on your tax return.