Who reimburses tax refunds for mileage?

One or the other of us is very confused.

There is no "tax refund for mileage" (at least in the US; I can't imagine there would be anywhere else, either).

You can, if you itemize deductions on your taxes, include a mileage deduction for miles driven for business purposes (you can only count mileage in excess of that you normally drive to get to work each day, so if your office is 20 miles away, and you drove to a client's office 30 miles from your house instead, you'd only get to deduct the extra 20 miles - ten each way - you drove to get to and from the client's office instead of your own). Also, if your company has already reimbursed you for the mileage, you don't get to "double dip" and take the deduction as well (though you also don't have to count the reimbursement as income, as long as it wasn't in excess of the IRS-specified amount per mile).

Since it's not actually a refund (though it can impact your refund amount), nobody "reimburses" it... it's effectively included along with all other credits towards taxes paid.