The Loyal Orange Institution is a vehemently anti-Catholic Protestant organisation loyal to the British Crown and dedicated to the continuance of direct rule from Britain in Northern Ireland, hence the word “loyal”.
The word “orange” refers to King William III, Prince of Orange, who was responsible for victory at the Battle of the Boyne in 1690.
The Battle of the Boyne was won by William of Orange and it was the turning point in King James’ unsuccessful attempt to regain the crown and ultimately helped ensure the continuation of Protestant supremacy in Ireland.
It is organised on strict Masonic lines and is self governing. It probably comes under the ban directly, however most, if not all of its adult male members are also members of the Freemasonry, including the “Royal Arch”, banned by the Catholic Church and others.
Current list of those Faiths and Churches that have either banned or condemned Freemasonry:
The entire world of Islam
Celtic Christian Church
Roman Catholic Church
Synod Anglican Church of England
Synod Australian Anglican Church
Methodist Church of England
General Association of Regular Baptist Churches
Orthodox Presbyterian Church
Presbyterian Church of Ireland
Reformed Presbyterian Church
Presbyterian Church in America
Free Presbyterian Church of Scotland
Wesleyan Methodist Church
Church of Scotland
Baptist Union of Scotland
Free Church of Scotland
Russian Orthodox Church
Assemblies of God
Church of the Nazarene
Evangelical Mennonite Church
Christian Reformed Church in America
Independent Fundamentalist Churches of America
The Evangelical Lutheran Synod
Lutheran Church Missouri Synod
Wisconsin Evangelical Lutheran Synod
I hope this helps.