It isn't quite as simple as licensed/unlicensed as being the deciding factor as to whether an ATZ exists.
An unlicensed aerodrome having an ATC unit (not sure if there is such a thing) or FISO unit where the hours of watch are notified in the AIP would have an ATZ during those hours. Also a normally licensed aerodrome with a notified ATC unit that becomes temporarily unlicensed, say because of no fire cover, would not lose it's ATZ status.
On the other hand, a licensed aerodrome which doesn't have a regularly manned RT frequency, by which I mean the hours are notified, would not have an ATZ.
See Rule 39 of the Rules of the Air.
Old Warden is occasionally licensed to permit pleasure flights by fare paying public but because the airfield isn't published in the AIP and does not have notified hours of watch of the FISO unit then it does not have an ATZ, even on display days. Displays are notfied by NOTAM but this does not confer ATZ status. We are trying to do something about this.