Based on the rather short wiki article on them, the Anglo-Irish seem to have been a prominent community before independence. However, I'm struggling to find any sources on precisely what happened to them after the formation of the Republic of Ireland. Did they "return" to England or remain in Ireland? If they stayed, did they retain any distinguishing cultural practices or were they simply absorbed into the Irish mainstream? Can one still find wealthy, RP-speaking people who closely identify with England in Dublin?
Also what happened to the noble titles many of these families had? Are they still "officially" recognized in the modern republic? In Britain?
Conveniently I did a Master's thesis on these people. It depended on how diverse their assets were. Irish based aristocrats with lands and financial interests in Britain did usually go there. Others remained in Ireland. Early on some were involved in Free State politics through the Senate but they steadily dropped out.
Titles held under British law continued but had no status in the Republic. Confusingly there were two kinds of peerage (Lords), UK peers had seats in the British House of Lords by right and retained them, but Irish peers (pre 1800 titles) had elected a group of 'representative peers' and these were discontinued.
Economically, the land reforms of the British era were completed by the new state so rental income became less relevant. The great country houses became more of a burden than an asset and many were sold off to pay debts. Contrary to popular myth, not that many were burned by the IRA. Some families did reinvent these estates as tourist destinations. The most famous example is Lord Henry Mount Charles who has become a successful concert promoter using Slane Castle. Powerscourt House is another good example.
There was a strong Southern Unionist identity among the first generation of aristocrats but subsequent generations assimilated. The Protestant population was too small to survive outside the mainstream and while Ireland was a firmly Catholic country, there were no attempts to tear down Protestant assets or business interests so the community survived.
In Northern Ireland some aristocrats remained in political life, notably Lord Londonderry who served as Minister of Education. Peerages are still awarded to Northern Irish people in the honours lists.