Ken Levine, a former "Cheers" writer, looks back at some of the contrivances on the show. But I can say right off the top of my head that a lot of people paid for drinks on the show. Norm also had a "bar tab" that was a 400-page binder.
Still, a lot of paradoxes on that show (as pointed out by Levine and commenters): everyone drank all day every day but no one ever was drunk. Carla was the only waitress, at times no one was ever tending bar, and Cheers had remarkably few employees for such a large bar. Cliff wears his postal uniform at all times, even on Sundays. Elite professional Frasier is both friends with the rest of the gang, and spends as much time at the bar as the unemployed Norm does. And I don't remember anyone ever ordering anything but beer.
Since I probably saw 200 episodes of Cheers before ever actually setting foot in a bar, none of this occurred to me as a kid. See also: one of my favorite SI pieces ever, by the great Steve Rushin.