All day free refills in the parks? Not that I know of. There are a few QS restaurants that have self-serve drink stations where you could get a refill during the time of your meal. Restaurantosaurus comes to mind at AK, Backlot Express at DHS, Electric Umbrella and Sunshine Seasons at Epcot, EoS and WPE at DTD.
If you have a refillable mug from your resort, you can get unlimited refills all day and all night at your resort, for the price of one mug (around $14).
AK has a single-day sports bottle to purchase with refills for $1, but I wouldn't guess this would be for coffee.
The water parks have an all-day refillable mug. Don't know if coffee works with these mugs.
There is a list of real coffee at the parks at WDW:
http://allears.net/din/coffee.htm
I like my real (not machine mix) iced latte from the Joffrey's stands around the parks. They are around $4 and can be bought with snack credits from the dining plan. At AK, no lid or straw is allowed because of animal safety. I bring my own iced drink container (from Starbucks) with locking lid and locking permanent straw, and AK is OK with that.