Why do some people write these differently? And does it mean "Please close the door" or "Please keep the door closed"?
"Türe" is an old-fashioned variation, "Tür" is standard today (and usage of "Türe" is fading away progressively in my experience. I haven't seen it in a while now).
In both cases, the correct translation is "Please close the door". If you want to say "Please keep the door closed", it's "Bitte [lass/lassen Sie] die Tür(e) geschlossen".