The seller of the home or property should pay the closing costs from the start of the real estate tax year until the actual closing date.
So the seller should be responsible for paying the taxes owed before closing and the buyer should pay real estate taxes on the property after the closing date.
You may however have to reimburse the seller for property taxes paid that started during the date of the sale until the end of the tax period.
So you may or may not have to pay any taxes during the closing process but do be prepared to pay the property tax just in case.