You can do as much refi as you like, there is no legal limit. But you do have the extra accounting matter that can be very complicated.
The closing cost of a refi, includes many different types of charges. FIrst are financing chargs such as appraisal, origination fee, title insurance etc. These are added to your cost basis for the house, and can be deducted when you sell the house. These costs cannot be deducted against your current year rental income.
Other part of the closing costs, such as prepaid interest, prepaid monthly payment, prepaid tax/insurance, are counted as expense of current year, and can be deducted from rental income.
A better answer might be here: https://ttlc.intuit.com/community/investments-and-rental-properties/discussion/what-closing-costs-from-the-rental-property-i-purchased-can-i-deduct/00/767375 But even that might not be exhausitive. Your best bet is to understand the theory and do your own homework.