Can I invest in shares in Australia and avoid being subject to Irish taxation?

Your charge to Irish CGT depends on your Irish tax residency status. Note that tax residency does not simply mean living in Ireland.

If you are resident or ordinarily resident in Ireland, you will be subject to Irish CGT on your worldwide gains (e.g. Australian share gains). You are resident if you have spent more than 183 days in Ireland, or more than 280 days over a 2 year period (with at least 30 days in each) in Ireland. You become ordinarily resident if you have been resident for 3 consecutive years in Ireland, and will remain ordinarily resident for the 3 consecutive years after you lose your residence. So for example, if you moved to Ireland in January 2020 and stayed here until January 2023, you would be resident for each of the years 2020, 2021 and 2022, and ordinarily resident for the years 2023, 2024 and 2025. In this situation, any disposals of any shares anywhere in the world would be subject to Irish CGT for the years 2020-2025, notwithstanding the fact you left Ireland in 2023.

You may simultaneously have a charge to Australian CGT on the same disposal, depending on the rules there which I'm not familiar with. However, as Ireland and Australia have a double tax treaty you will not pay tax twice on the same income. If you do pay Australian CGT, you will get a credit for this against your Irish CGT liability (if applicable). You will still pay the 33% rate of Irish CGT, but get a credit for the (lower) amount of Australian tax you paid. You will only lose your liability to Irish tax once you're no longer resident or ordinarily resident, as described above.

As a general rule dividends received are subject to income tax (instead of CGT), and this applies regardless of whether they decide to reinvest the proceeds or not. Their liability to Irish income tax is again based on whether they are resident or ordinarily resident, i.e. taxed on worldwide dividends if resident or ordinarily resident, otherwise not. And again, a credit for Australian tax should be available under the Irish-Australia double tax treaty so the same income is not taxed twice.