It depends. Refinancing may or may not affect your taxes, it depends on what type of refinance you did and how you file. Generally, your mortgage only impacts your taxes if you itemize your deductions.
There are caps on the amount of deductible interest that you may deduct as an itemized deduction. Interest can only be deducted on mortgages up to $750,000 ($375,000 in you use the MFS filing status) so if your loan was more the cap, your deductible interest will have to be limited. For a cash out refinance, you must use the cash out to make capital improvements (permanent addition that increase the value) to your principal residence or second home.
You may also deduct points (prepaid interest). Other names for points are loan origination fee, loan discounts, or discount points.
For more information, please review the TurboTax articles Mortgage Refinance Tax Deductions and Home Improvements and Your Taxes.