It's just as if you'd bought a car and had a car loan. Then, after a few years you decided you wanted a new one, so you bought a new car and your wife ended up driving your first one. You wouldn't have to go back and refinance or get a new car loan on that first car.
So, don't worry about that.
As for refinancing, now you run into the difference between rental properties and primary residences. And, yes, it's likely that you'll have to pay more, overall, to refinance the rental than if it were your primary residence. However, check with a good mortgage broker for guidance on that.
Hope that helps.