What refinance cost? What tax deduction?
I will comment on two issues you might be asking about.
1. Depreciation.
When you place the house in service as a rental, you must list the value for depreciation. You depreciate the property over 27.5 years. The larger the value you can list for depreciation, the more of a deduction you take each year.
The value you use for depreciation is either:
a. your adjusted cost basis, or
b. the present fair market value, whichever is less,
Then you subtract the cost of the land (since land does not depreciate).
Adjusted cost basis is the original purchase price, plus the value of permanent improvements (but not repairs). Some of the closing costs associated with the original purchase and the refinancing can be added to the adjusted cost basis, but probably not all the closing costs. Items that can be used to adjust the cost basis are listed in publication 523 on page 8.
https://www.irs.gov/forms-pubs/about-publication-523
For example, suppose the house cost $100,000, your original closing costs were $3000, closing costs on the refinance were $4000, and the portion of the original purchase price attributed to the land was $15,000. The basis you list for depreciation is ($100,000 minus $15,000 plus $7,000)= $92,000.
2. Mortgage Interest expenses
Your monthly mortgage interest expense is one of the expenses you can deduct from your rental income. You can also deduct insurance, utilities, and property taxes that you actually pay. You can't deduct your entire mortgage payment, because you can't deduct the principal part of the loan and you can't deduct escrow amounts (only the amounts actually paid to the insurance company, property tax department, etc.). You can deduct mortgage interest on a property that you rent out even if the mortgage is more than $750,000 (which makes it not fully deductible on schedule A if it was still your personal home).
If you are asking a different question, please provide more details.
