Every metric has 2 sides. refinancing at a higher rate WILL technically reduce your cash flow (though as Jaysen Medhurst suggested, not much), but consider what else rising rates will do. Rising rates are a sign of impending inflation. Inflation is FANTASTIC for those who own hard assets like real estate, as their principal owned remains the same, yet the rents should climb in correlation with inflation rates. In the extreme case of hyperinflation, for example, the building you bought for $1M and originally collected $10K/mth on will still have the same mortgage payment, but you can theoretically collect $20K/mth in rent. Other expenses will climb, but that major debt piece remains, and improves your cash flow.
Additionally, if you're looking at multifamily, as interest rates increase, so should the expected cap rates, which will reduce the market value of a property (again, in theory).
With all the quantitative easing that occurred in the last decade, the US is due for some SIGNIFICANT inflation in our near future, so I'm intending to buy as many assets as I can.