Although of course it's not required, it would be customary for you to tip the driver. Just because the service is free should have NOTHING to do with it. Think of it this way. Let's say someone bought you a restaurant gift certificate for your birthday that covered the cost of your meal. After the waitress brought you your check, would you simply give her the gift certificate or would you also tip her based on the retail cost of the meal?
Disney DOES pay the tips of Bell Services CMs for getting your checked luggage to "magically" appear in your hotel room, but the DME drivers, employees of a subcontractor, do not receive any "tip" money from Disney. AGain, it's a matter of personal choice, it's not a requirement, but IMHO I think tipping the DME driver for even adequate service is quite appropriate.
-- Eric