What would be the point? Even aside from the fact that it would inevitably be seen as highly provocative, there are absolutely no advantages in such membership.

There are of course two countries in the Commonwealth that never had any coneecttion with the British Empire, Mozambique and Rwanda, but for them it was a matter of building up links with close neighbours, such as Kenya and South Africa. For Ireland there are already close links with neighbours, all of which are in the EU. Even the UK still is at this point, and there's still time to preserve that.

If the EU decided to set up a formal link with the Commonwealth, or the other way round, I don't imagine Ireland would have many issues with that. Those would be the only circumstances in which it would be conceivable for Ireland to belong.