Can a lender on an auto loan require a co-borrower or guarantor to go on the title?

The lender can require just about anything, but it is more likely that they will want every name on the title to be on the loan, not the other way around. If anyone should insist the cosigner's name be on the title, it would be the cosigner himself. That will give him a right to take possession of the vehicle if he is stuck with the payments.