I’ve decided to give this question a separate answer in light of the fact that there seems to be some confusion still, possibly arising from the fact that people are mixing information between naturalising as an Austrian and naturalising with another EU citizenship that allows British citizens to retain their British citizenship as well as taking another EU citizenship for which they are also eligible.
The answer is simple. If you take Austrian citizenship you no longer have either the Anmeldebescheinigung (this is part of ensuring you this ability to exercise the freedom of movement), which you get after three months of living in Austria. If you have done a Bescheinigung des Daueraufenthalts you turn in this form in any case. If you naturalise as an Austrian you surrender whichever of these you hold when you become an Austrian citizen (for pre 2006 arrivals your Meldezettel or Bestätigung der Meldung covers the Anmeldebescheinigung).
If you are taking an additional citizenship of another EU country to supplement your British citizenship, the advice is to update your registration to reflect your new status i.e. amending your current residence documents to reflect your supplementary nationality.
This advice was confirmed by the HM Embassy in consultation with the Innenministerium as per the tweet exchange below.