Cheers was funny because it had a great ensemble cast and they played really well against each other. Basically, it was a show that ran on great character-based humor, a rare beast indeed. There are like two shows on television that still do this. Not that there were any more back then.
(It's Always Sunny in Philadelphia and Parks and Recreation, in case you were wondering.)
Wow, someone else who likes "Parks and Recreation" (or, as I call it, "The Office if Everyone Weren't Retarded").
"Cheers" was actually two shows in one. The 80s had lots of shows with male and female leads doing the flirty love/hate thing, and that's what drove "Cheers" for the first couple seasons, while the secondary characters slowly came into their own. After that the show relied more and more on the character-driven humor, and once Diane left, the show went 100% character-driven.