You got a couple different issues here, and some bad information too.
First, your sister owns the property, not the lender. The lender has a lien. When the property is transferred, the mortgage will have to be paid off as part of the transfer, and the lien will be released.
Whether or not an interest only loan is crappy depends on the situation. For most people, it's the wrong loan. There are some circumstances that are great for I/Os, but not for the average person.
"Your mortgage is underwater." Now, does that mean the loan on your sister's house is underwater or do you and your fiance own a different house? If you have a loan on another house and you're underwater on it, how are you going to get a mortgage to pay off the loan for the house you want to acquire? Somebody has to pay off your sister's loan before you can be the owner. If your sister's loan is underwater, that's why she can't refinance it. She would owe more than the value of the house. Investment property can be refinanced so if she's isn't underwater, there's probably some other reason you aren't aware of that prohibits the refi. Maybe she needs to try a different lender. Not all lenders deal with investment property the same way.
You can move into her house with her permission, after all it's rental property. Selling it to you is a possibility (if you aren't abandoning another mortgage...if you are doing that, no lender will give you another loan for a long long long time) and she can give you a gift of equity. If she gives you the house, the loan has to be paid. If she sells you the house, you would get a new mortgage in your name and that pays off the loan.
Can you get a mortgage? That's a big question here and I'm not clear on the details. I have a very literal mind, so when you say you're not on the warranty deed and then say our mortgage is underwater, you confuse me. I have to assume you have another house. She's stuck and our mortgage makes it sound like two different things.
No you can't just be added to the mortgage. There's a chance you could be added to the warranty deed but not be responsible for the mortgage. If I was your sister, I wouldn't like that. Without seeing the loan docs I don't know; it would be rare, but possible. Your sister could call the title company that prepared them and ask if granting part of her interest in the property to you would excelerate her loan, i.e. make it due. They probably can't talk to you about it.