I know that some key fighters for Irish independence were Anglican, and that Ireland has had an ambivalent relationship with Catholicism in modern times. That said, has there been any serious effort by Irish Catholic bishops to reclaim Church of Ireland properties like Christ Church Cathedral or St. Patrick's Cathedral?
This is a partial answer.
To some, this question may seem strange, but it needs to be understood in context of neighbouring countries. As @PeterTurner points out, the ancient cathedrals in France are state-owned, but are run entirely by the Roman Catholic Church and function in every way as any Roman Catholic cathedral does.
In England the ancient cathedrals are state-owned, via the national established church, the Church of England. This is because England is officially a Protestant country as His Majesty's Coronation reminded us recently.
So many people assume that the Dublin Cathedrals are state-owned, as their French and English counterparts are, and that, since Roman Catholicism is the predominant religion of the Republic of Ireland, it seems natural that the Roman Catholic church should run the cathedrals.
Dublin, unusually, had two cathedrals before the Reformation, both now Church of Ireland. There is also a Roman Catholic pro-Cathedral built in 1825. It is called a pro-Cathedral because, in the view of the Roman Catholic Church, Christ Church Cathedral is the real Roman Catholic Cathedral and proper seat of the Archbishop of Dublin. The pro-Cathedral functions exactly as a cathedral.
Many tourists assume that Christ Church and St Patrick's Catherals are Roman Catholic and that the pro in pro-cathedral is an abbreviation for Protestant (on tourist maps etc.)
However, since 1871, Dublin's two historic Cathedral's have not been state-owned. From 1st January 1871 the Church of Ireland was disestablished and disendowed. The United Church of England and Ireland was separated into two parts. The Church of England remained, and remains, the established and official state church in England. The Church of Ireland was constituted at that time as an independent denomination. All buildings in Ireland in actual use by the United Church of England and Ireland at that date passed to the new denomination which was constituted as any other private organisation. Land and endowments not in actual use for religious or educational purposes remained state-owned and were used in various ways, but mainly to fund education.
So by the time the Irish Free State was formed in 1922 Dublin's two ancient cathedrals were effectively the private property of an independent organisation, the Church of Ireland. As they were no longer state-owned, the Irish government had, and has, no say in what happened to them.
This is only a partial answer because it could be argued that the Republic of Ireland government could compulsorily take over the cathedrals despite their being, in a sense, private property. I am fairly certain that this was never seriously considered.
It is possible that improved eccumenical relationships, and/or maintenace costs, may at some point bring about changes.