If the water dispenser is in a boat-house, crawl-space, outdoors (generally), Garage (generally), Bathroom, unfinished basement, or within 6 feet of a sink in a laundry room, or fed from a kitchen counter-top receptacle, yes.
A piece of equipment (like your water cooler or a blender in the kitchen or a hair dryer in the bathroom) is never required to have GFCI protection. The thing that requires GFCI protection is the receptacle.