So… context required.
In 1998, the Good Friday Agreement was signed in an effort to bring to an end the Troubles - 30 years of political/religious conflict - in Northern Ireland. Among the bits and pieces of a complex agreement was that people born in Northern Ireland (perhaps only the Northern Irish, not wholly clear on it personally) could consider themselves Irish, British, or both. This extends to sports, and international competition.
The Commonwealth Games are similar to the Olympics, in that it has similar events and happens every four years, but is open only to members of the
Commonwealth of Nations. One quirk of this is that the nations making up the United Kingdom - England, Scotland, Wales, Northern Ireland - compete separately. Ireland is not a member of the Commonwealth, hasn’t been since 1949.
Even if someone from Northern Ireland competes for Ireland in other competitions (world or European championships, the Olympics) they have been able to represent Northern Ireland in the Commonwealth Games, because NI competes as part of the British team in other competitions.
The International Gymnastics Federation (FIG) has taken it upon themselves to ignore a landmark agreement and a quarter century of precedent to say that NI gymnasts that have represented Ireland cannot represent NI at the Commonwealth games. Their solution is that the affected athletes either change the nationality on their FIG registration, or the games themselves remove the gymnastics as an FIG registered competition. Because apparently they know better how to handle sensitive topics like national identity that cause people to kill each other for decades.
TL;DR the Northern Irish are allowed to consider themselves Irish after 30 years of killing, but not according to world gymnastics bureaucrats.