Whether or not a contract is needed (or advisable) is going to depend on the agreement/relationship with that 3rd party who is being bought out. Not a bad idea to get the agreement regarding the transfer of his ownership interest in the property in writing now so expectations are very clear.
At the very least that party is going to have to sign a deed conveying his interest in the property over to OP and his wife at the closing of the refinance. Sounds like money is going to change hands between OP and the 3rd party as well, and if they are wanting the title company to handle that in escrow along with the refi, title company may want a contract. Again whether that level of formality is necessary or advisable really depends on the relationship of OP and the 3rd party, but it's never a bad idea to be too formal, even when dealing with friends/family (sometimes especially when dealing with them).