Heating in Tasmania is either electricity or wood fire. Wood fire usage is much lower than it used to be as the city I live in the past has given generous grants to replace wood heater with electricity to combat air pollution (it's a valley so all the smoke used to just sit there).
So it's electricy only. In regards to heating most of Australia is either either using electric space heaters or reverse-cycle air-conditioning.
Tasmania is probably the second coldest place in Australia (after the Snowy mountains). Some cursory googling suggest that even there central heating is not common.
(Natural Gas in Australia isn't that cheap anyway, we export nearly all of ours)
We have gas heating, floor heating , wall heaters and such..
However the weather doesn't get cold enough to kill you here even down south. The houses generally aren't designed for cold weather and we as a nation tend to just put on more clothes and endure somewhat colder houses. I've had people from overseas complain about how cold they get because we don't do much about it.
In say, Germany/China in winter I remember being stunned by how hot people kept their houses and public spaces. We don't heat our spaces in the same way generally.
On the other hand, it gets real hot and we are somewhat optimised towards that way of living.
I do not know how far is this from Zero CO2 considering cattle production and CH4 emission, cement production, ship and airplane traffic ...