I do think the "accountability" rule applies to charities (as I was a treasurer of one) insofar as determining whether money paid to staff (including volunteers) is income or reimbursement. Someone who was paid (let's say) $50 per day to drive patients, without any accountability, would probably be considered to be receiving compensation subject to a 1099. By taking receipts and only reimbursing for actual mileage, and as long as the reimbursement is not excessive (the inurement principle) the charity avoids this problem and the money is not taxable income to the volunteer.
If the mileage reimbursement was more than the federal rate, the difference might be taxable income. If the mileage is less than the federal rate, it's just a non-taxable reimbursement.
