In DS in addition to WPE , Cooke's of Dublin also does refills. At WPE the staff has stopped at our table and offered to get DH his refill ( I'm having an adult beverage)
Blaze pizza in Disney Springs has free refills on sodas and their specialty Aguas Frescas and Lemonades. Also very good pizza. For me and my better half, this is now the *best* quick service option in all of Disney World.
I honestly don't remember. I just remember taking it to the counter and asking the CM "Can I please get some more Diet Coke?" and she gave it to me, but I don't remember if she refilled my cup or just handed me a new one.