at Table service, if you're ordering soda fountain beverages (Coke products), or tea or coffee, you will get unlimited refills.
However, some table service restaurants allow bottled water, smoothies, milkshakes or "virgin" beverages like pina colava without the rum. Usually if you order one of those specialty drinks, it's one serving only, no refills. Course, those are usually large drinks...like the milkshakes at The Wave were probably 16 oz. shakes. You can get ice water with those specialty beverages too.
At counter service in the resorts, most of them have self-serve soda fountain...so of course you can refill your beverage as much as you like. If you opt for a bottled drink, you'll only get one. In the parks, some counter service places have self-serve soda fountains (Like Sunshine Seasons in Epcot) where you can refill your glass if you like. But the places that are straight counter-service where they hand you the beverage, no refills.