I'd pronounce it basically the same as "June", but if I was trying to pronounce it "properly" it would be more like "dewn", where dew rhymes with jew. Sounds very weird to my ear to hear it rhymed with moon.
They do rhyme, it's the bit at the start that's different. Dune is sort of "dyoon", not "doon". I should really learn the international phonetic alphabet to actually explain this properly.
Apparently both pronunciations are correct but I've never heard it pronounced any other way than doōn. But I think what is required for the joke to work is to pronounce doin' like doōn.