What you're talking about is called the "harp quarter" of the monarch's coat of arms, and it certainly represents the monarch's claim to Ireland, but currently the legal claim extends only to Northern Ireland. There's no intent by having the symbol there to claim the Irish free state.

Over time, the royal coat of arms has definitely changed to reflect claims to territory or lost claims to territory -- for example, Edward III quartered the lions of England with the fleur-de-lis to symbolize his claim to the French throne, and the French symbols stayed there until 1801, except for a brief change during the English Commonwealth when there were no royal arms. The royal arms dropped the French symbol when the Act of Union formed the UK and the monarch dropped his claim to the French kingdom (France had, of course, become a republic in the meantime).

Now, in a theoretical future world where Ireland is reunified as an independent state, would that quarter be dropped from the Coat of Arms? I would assume that's a likely outcome, since the UK would then revert back to being a confederacy of England and Scotland. There's no historical precedent for what those arms would look like, as James I of Scotland inherited the Irish crown as well as the English in 1603, and quartered the harp into his coat of arms.