Co-Borrower does not want to proceed?

I want to know what the best practice is for a situation that came up. We are refinancing a loan for three borrowers, but during the loan process the third borrower decided he did not want to continue with the loan.

Can this loan proceed with the two borrowers and new earlies issued in just their name and document what happen or do we need to close this file and start over with the other two borrowers?

I don't think you need to do anything. Just take the name off and print the final docs in just the two remaining names. Depending upon the collateral, if any, if the borrower who wants out owns all or part, that borrower will still have to be a grantor. You said "refinance." Is the bank OK with letting the the third borrower off the loan?

When the loan that is being paid off originated the third borrower was necessary to qualify, but now the two borrowers qualify fine on their own and the third borrower is not needed; therefore, he wants to be removed

In the banks I've worked at there was a general rule that a person could be added to an application, but the loan would be decisioned (usually as withdrawn or ABNA) and a new app would be required when someone was dropping off. Just to make the decisioning process a bit cleaner.