I was born and live in in Australia, but have Irish citizenship also because my father was born in Ireland. I have an Irish passport but not an Australian one. I want to visit the Cook Islands later this year. Can I fly out of Australia for two weeks and return using my Irish passport? Or, do I have to get an Australian passport?
Australian citizens entering and leaving Australia
Australian citizens should use their Australian passport to enter and leave Australia.
If you have another passport, you may use that after you leave Australia. A passport is the best way to show that you are an Australian citizen. You may still be able to enter Australia if you are a citizen without a passport, but it will be more difficult. The airline may stop you from boarding a plane to Australia. Smart Traveller has more information about travelling as a dual citizen.
I take this to mean that an Australian citizen not carrying an Austrialian passport will not be denied entry, but won't be allowed through Immigration until citizenship status is established.
Your Irish passport might give you the ability to enter Australia without a visa, or by obtaining a visa upon entry. However, given the possibility that the airline may not board you for an Australian-bound flight, obtaining an Australian passport and using it seems an easier and less fraught path.
————————- Edit January 23:
@phoog points out in his comment that an Irish passport holder without an Australian visa will not be allowed to board an Australia-bound flight.