At Hollywood Studios, we went to the Tune In Lounge which is the bar of 50's Primetime Cafe. We didn't have any ADRs, we just went into the bar/lounge area and ordered drinks.
At Downtown Disney, we were able to walk right into T-Rex and sit at the bar and order drinks. Again, we didn't have any ADRs.
At Epcot, we ordered drinks from the Rose and Crown Pub without having ADRs for the restaurant part of the pub.
Those are the only restaurants I specifically tried doing that at, and they both allowed it. I'm sure other restaurants allow it to... those are just the only ones I tried.
As far as other places to just order drinks...
At Animal Kingdom you can get drinks at the Dawa Bar. Also near the Yak and Yeti CS restaurant there's Yak and Yeti Quality Beverages where you can order alcoholic beverages to go.
At Hollywood Studios there's the Backlot Canteen next to the Studio Catering Company that sells alcoholic drinks.
At Epcot there are PLENTY of places to get drinks in the World Showcase. (La Cava del Tequila in Mexico is a personal favorite of mine!)
And many of the moderate and deluxe resorts have bars and lounges in them where you can stop in and get a drink. And I think all the resorts have a pool bar where you can order drinks.
Downtown Disney also has a lot of bars, clubs and lounges to get drinks.