In this situation, the funds were paid to a 3rd party mover, however, it was a necessary expense in order to make the donation so I would be more inclined to argue that you could deduct the expense.  

To your point, if you had to drive the item over to goodwill (oh wherever) yourself, you could deduct the mileage so I don't see why you can't deduct other costs associated with the transportation of the item.