Tone is hailed as the father of Irish Republicanism, so it seems that any organization interested in that goal would have to, on some level, engage with his legacy. Given the fact he was a Protestant though, and that Republicanism was nearly synonymous with Catholicism with at least some factions during the 20th century, how did this affect the remembrance of Tone by militant, and Catholic, Republicans?
Additionally, there being several "IRAs", I'm not too picky, and would love to see how the views changed from the Volunteers/Republican Army to the anti-treaty descended IRA, and of course, the PIRA perhaps most of all.
I will try to answer this to the best of my ability, but becasue of the vast amount of republican, nationalist and catholic groups out there it may be hard to talk about each on it's own, though i do believe they can all be covered broadly. Even if we just pick the ones with IRA in their name, that's still a lot of groups with over 100 years of history. Then there are still more groups from Ireland, rethought the War for Independence, Civil War, Border Campaign, Troubles and the current conflict who fought against the British but don't have the name "IRA". Groups like the IRB, INLA, RAD would still be considered the "IRA" by many but all three fought in different wars, and are separate groups than the IRA's. For many nationalists and republicans, it's seen that the Troubles in constantly painted as a religious war, as to discredit the cause.
Now that that's out of the way, Wolfe tone was one of the leaders of the rebellion of 1798. This is a very important date for the Republican and nationalist communities, leaving Wolfe tone, as you called it, "the father of Irish Republicanism." Almost everyone in Ireland knows his name and knows who he is. The famous Irish Republican band, "The Wolfetones" are named after him, and are responsible for many of the Irish Rebel songs, praising the IRA, Wolfe tone himself and other protestant men who fought for Ireland. One being Protestant Men. For the Society of United Irishmen, most of their founders and leaders were Presbyterians. Although the United Irish men, had a lot of men knows as "defenders" in it's ranks, who were involved in religious conflicts against protestants, protestants and Catholics fought along side one another.
The next significant uprising in Ireland was the Easter rising. Involving the Irish Volunteers, Irish Citizen Army, Na Fianna Eireann, Cumann na mBanand the Hibernian Rifles. Na Fianna Eireann were a boy scouts organisation, with the intention to teach the Irish youth about the History of Ireland, the Irish language and to train them for future Irish rebellions. Their founder, Bulmer Hobson was a Quaker. Roger Casement, another protestant traveled to Germany to gather weapons, support and troops. Many more protestants fought in Dublin over the rising. The struggle in Ireland for independence wasn't a Catholic one, without the help from protestants, the organizing needed for the different groups wouldn't have been done. Many of these men also fought during the war of independence.
The GAA, the organisation in charge of the Irish games like hurling and Gaelic football, is seen as a republican organisation, and which have a lot of stadiums and pitches names after protestant men. The largest stadium in Northern Ireland is named after Roger Casement .
After the war of independence, many more Protestants joined Republican groups. During the border campaign and the troubles, there were more protestant men who fought and died with groups like the INLA and the PIRA. The PIRA with 30,000 members, were the second largest Irish Republican militant force after the Irish volunteers back before 1916 with around 180,000 men. These men would look back upon the leaders of 1978 and 1916 as heroes. There are still marches an commemoration for Wolfetone. This one for example is a march of Republican bands, marching to his grave, playing Irish republican rebel songs.
To sum up, the conflict was never really a religious one, the two sides of the conflict are the native Irish and the descendants of Scottish and English planters in Ireland. One being staunchly catholic, the other being staunchly Protestant. They fought becasue of the opinion of the freedom of Ireland and how it's people should be treated. There is nothing to stop a Protestant or Planter from believing a free united Ireland is the best choice. The religion of Wolfetone was never a problem to any true republican or nationalist. Many nationalists, like myself, find it upsetting to see that the conflicts throughout Ireland's history be painted as religious conflicts, rather than those of freedom and civil rights. I know you said that you wished to see how views changed from organisation, but I have never noticed any. Likewise, i have never known or heard of anyone who dismissed Wolfetone as he was Protestant.
Well the first thing to be said is that the notion that there are ‘Irish Catholic” groups should be put aside and regarded instead as ‘Irish nationalist’ or ‘Irish republican’ groups. The IRA is the best known but when it emerges in the early 20th century the IRA actually incorporated many other groups such as the Citizen Army, the IRB [Irish Republican Brotherhood] and the Irish Volunteers who all fought in the 1916 Rising. These groups were not looking for rights regarding religion – they were formed to bring about the independence of Ireland from British rule. Wolfe Tone is considered a great hero to all these Irish independence groups. He was one of the leaders of the 1798 rebellion and as such he is indeed considered to be the ‘father’ of Irish republicanism. Both the IRA and Sinn Féin hold yearly commemorations in November at his grave in Bodenstown in Kildare on the anniversary of his death.
Your question brings up some very broad issues and I will try to give you a concise but lucid explanation of the history of it all. The question of Irish dissatisfaction with English, and later British, rule goes back over 800 years so it’s not possible to give the entire history here but for the sake of emphasis that the conflict is not religious based let me state the following. There has been conflict about the position of Ireland in relation to England going back to the first incursions in the twelfth century. That is, centuries before the Reformation. To give you some examples of what the issues were - many of the earliest conflicts beginning in the 12th century were about land ownership as native Irish were forcibly pushed off traditional lands and ownership given to the new Anglo-Norman settlers.
Cultural hegemony was an important part of that colonising plan. These new English settlers were to segregate themselves from the native Irish population. The Statutes of Kilkenny in 1366 laid out the ‘regulations’ as regards the position of the English landowners vs the native Irish. The ruling class, the English settlers, were not to adopt the Irish language or use it in anyway. English was to be the language of the colony. The Irish Brehon law system was outlawed and English common law was to be established but rights under English law were confined to those within the English settlements and not to the native Irish. There was to be no marriage between the two communities, in fact the statutes stated there was to be no sexual relations at all. Irish poets and bards were not allowed near any English settlement or English homes. The Irish game of hurling was not to be played by the English settlers. The purpose of these laws was to create an upper ruling class who would dominate and not be corrupted by any native aspirations or interests.
It is important to note however, that these regulations failed and the Anglo-Norman families adopted Irish ways including the Irish language and the Brehon law system. But the intent is clear and the native Irish were not given any rights under the statutes. By the year 1400 the English control and influence was limited to an area around Dublin, which became known as The Pale. Beyond the Pale the native Gaelic Irish continued to live a fairly autonomous life using the Irish language, the Irish law system and Irish cultural forms. Raids by the Gaelic Irish on the Pale area were common.
Throughout the next 100 years the Gaelic Irish continued to rebel against any English incursions into native lands. The Norman settlers had intermarried into the Irish and became almost indistinguishable from the native Irish. By the late 1400s the Irish colony was giving much trouble to the English administration as the Parliament in Dublin – under the control of the now Hibernised Norman families - strove to make laws contrary to the wishes of the overseers in London. In 1494 the English passed a law which became known as Poynings Law which curtailed the power of the Irish parliament and essentially put it under the control of Westminster. Unrest in Ireland continued.
By the time the Reformation came around some 40 years later there was no appetite in Ireland for a new English religion. Irish resistance to the Reformation - by both the Gaelic Irish and the Hiberno-Norman settlers - had more to do with disgust at the English than any religious fervour. The Irish simply did not want any more English interference in Irish affairs. Religion was used as a way of not conforming to English rule. Religion then became another 'tool' that the British could use to identify the native Irish and possibly eliminate them for being a threat to their governance.
So in response to this resistance in the early 1600s the English decided on a drastic plan to dispossess the native Catholic Irish and sent in Protestant 'planters' – mostly Presbyterian - into the Ulster region to replace the native Irish. At this time Ulster was the most Gaelic part of Ireland. This resulted in an economic class of Protestant owners in the region while the native Catholic Irish were dispossessed. This was the Plantation of Ulster the repercussions of which have existed down to the present day.
In the century following the Ulster Plantation Catholic Irish rights were further curtailed. The Penal Laws of the 18th century were aimed at eliminating all Irish Catholic economic and political power. Catholics were denied voting rights, rights to sit in Parliament [Dublin or Westminster] , or sit in local assembles, they had no rights to ownership of any property. Even Catholic marriage was illegal. Tenant rights for Catholics - for the whole of Ireland - were dismal compared to rights granted to Protestant settlers. The list of curtailment of Catholic rights is long.
To come to Wolfe Tone - over the years many Protestants in Ireland protested at the treatment of Irish Catholics by the British government. Some joined in with Irish Catholics in rebellion. Tone was not unique or alone in this. This extended to the constitutional path also. Charles Stewart Parnell, a Protestant Irishman, fought in the British House of Commons in the late 19th century for Home Rule for Ireland. The party that he was leader of, the Irish Party, consisted of Protestants and Catholics.
This is by way of explaining how the conflict is not a religious one - it's a deeply economic and political one and was fought by Irish nationalists on one side who decided that the best way forward for Ireland was to be independent of British rule. The various groups that you asked about, the Irish Volunteers, the IRA, the PIRA have all emerged as a result of this aim. And Wolfe Tone to all of them represents the embodiment of Irish republican hopes.