Domicile is one of those nebulous concepts which tax books don't tend to explain very well. Broadly, your domicile is that of the country where you were born. However, if you completely and irrevocably sever ALL links with that country and move elsewhere, you can assume a new domicile. In your own case, as Ireland and the UK have tax agreements with each other it probably wouldn't make a difference to your tax liability here ie the countries have rules setting out how earnings originating in one country are taxable in the other country. Similarly, if your earnings arise solely based on employment here, there are no attractive tax avoidance possibilities available to you due to the length of time you've been residing here.
Your domicile would possibly be important if, say, you happened to have earnings in a country outside of the EU. Depending on whether Ireland had a tax agreement with that other country, it might conceivably be possible to not pay any tax in Ireland on such earnings.
My guess- and I emphasise that it's a guess- is that you can tick the box to indicate that you are a 'non-dom' but it will have no real effect on your tax liability here.