I would love to learn your entire situation because mathematically it might work out for you to take a HUGE loss on your current home, just to get out of it, and realize faster gains through smart equity building on your next one.
Why refinance? If you did would it lower your payment significantly enough that the rent may cover it? Assuming you did this, do you have equity in the home? Can you afford the current mortgage and the new mortgage if your rental was vacant for a while?
Banks will typically use 75% of the rental income when calculating your debt ratio due to vacancies and maintenance. If your present income, plus 75% of the realistic rent would be enough to satisfy the bank then you possibly could, but you will need to meet with a local lender to make sure.