You are considered an employee of the S-Corporation and if you treat the mileage and transportation costs as a personal expense, tyu won't be able to take the expense on your 1040 return since employee business are no longer deductible.
Alternatively, the S-Corp could reimburse you and deduct the reimbursement. One thing that you can't do is let the S-Corporation claim the expenses themselves, as the S-Corporation doesn't own the car.
I have a similar issue and am having trouble with the balance sheet in turbo tax adding up. First question, for the answer just given where does the S-corp reimbursement to the "employee" (the owner-employee) get logged? Is it a deduction "office expense" or other or ???