We are unaware of any liability associated with mileage reimbursements. Allowable mileage reimbursement rates are set by the Internal Revenue Service, which increased the rate allowed for 2015 (see TAPN’s 2015 IRS Mileage Rates). Companies are free to match the IRS rate, pay less than the IRS, or pay more than the IRS rate; though if a company reimburses employees at a higher rate that the IRS rate, the overage must be reported as income to the employees.
Liability for use of a personal vehicle for company business is not related to the rate of reimbursement. The question of liability of business use of a personal vehicle has not come up before. In our experience, use of one’s car for any purpose is covered by one’s own insurance. We recommend that you contact your legal department and inquire about any liability to your institution if employees use their personal vehicles for business use.
(The 2016 IRS mileage rates are discussed here.)