All the counter service locations in the Resorts have refills with the Resort mugs that you can purchase for $11.99 + tax. As for refills on paper cups there is no sign that says they are not allowed and I have seen guests refill cups, however a Cast Member at Gasparilla Grill and Games said there are no refills in paper cups at the Resorts. So I really don't know what the correct answer is. The counter service locations in Downtown Disney that I have been to (Earl of Sandwich and Wolfgang Pucks Express) my family and I have always got refills and were never told we were not allowed. One reason why those locations have refills is because you get the drinks yourself. However each counter service location in the parks that my family and I have been to, the Cast Members get the drinks for you so obviously there they do not give refills unless you wanted to purchase another drink.
For the table service restaurants and buffets refills are free, so all you need to do is ask your server for another drink and they will get it without any problems.