In addition to asking for cups of ice water at counter service locations, many table service restaurants have a cooler of water with cups sitting in the lobby, for the use of guests. You can walk in and grab a cup or two even if you are not checking in for that restaurant. I did that last week at Mama Melrose.
Also, a couple of counter service places have self-service drink stations, which have a water option you can use even if you are not making a purchase. Backlot Express in DHS, and Electric Umbrella and Sunshine Seasons at Epcot come to mind. BOG also has them, but it would be awkward at best and possibly not allowed at all to walk in to use them without having lunch or breakfast there.