If the Palestinians were mostly Irish Catholics who settled there after the famine

[Skill and Chance]

This is really weird, but that makes it really interesting.

Given cultural memory of the Crusades and that this migration is happening prior to the 20th century international human rights revolution, I'm not envisioning a peaceful coexistence between the Irish Catholic immigrants and Arab Muslims at all.  If we assume the Irish are the majority, I would have to assume they took the land by force and exiled the existing population. 

I am also wondering if this would trigger a generalized Christian revolt against Muslim rulers throughout the Middle East once it became clear the Irish were obviously winning?  In this scenario, Palestine ends up being a majority Oriental/Eastern Orthodox state with an Irish Catholic martial elite.  Probably due to sheer demographic weight of all the Christians in the Middle East who possibly can migrating there and the intermarriage that would follow, most of the Catholics have converted by the early 20th century (you saw something similar with the Arian Visigothic military elite in early Medieval Spain, who became culturally indistinguishable from Catholic Spainards after a couple of centuries).

Irish immigrants IRL generally flocked to multicultural cities and embraced industrialization, so I think they would be able to do a lot of good in the Middle East in the medium-long run.  Though it would have a Christian majority would expect Irish Palestine to become the go to refuge for anyone who in the Middle East who isn't Muslim, including Jews, Druze, Yazidi, and some even minority groups within Islam who are suppressed by the local governments.  There would be faster industrialization and larger cities earlier than IRL.

Assuming the world wars still occur in a recognizable way, they would obviously join the Entente in WWI in opposition to the Ottomans.  The threat on their immediate borders would swamp their historical animosity toward England, as we saw with France IRL.  At the end of the day, the enemy of my enemy is my friend.  Military prowess has always been a huge part of Irish and Scots-Irish culture, with many of America's most famous generals coming from those immigrants, so I imagine they would fare quite well.  The extreme scenario of a total victory and Christian conquest of Turkey isn't out of the question, particularly if they are ahead of the curve in industrializing.  What is out of the question IMO is Britain taking over after WWI.  They would be fighting in the same coalition, after all.  Unless it's somehow a total loss to the Ottomans and Muslim reconquest coupled with an Entente victory overall later in the war, that just wouldn't happen.

The better they fare in WWI, the more likely they are to get involved in WWII and sooner.  In the extreme scenario where Emperor Sean I Fitzgerald of Constantinople has united the eastern Mediterranean under a multiethnic constitutional democracy, they probably end up in an early naval conflict with Mussolini's Italy.  If they lost outright in WWI, probably nobody cares until the Germans go looking for oil.  If the Irish-descended Mediterranean state becomes a world power, it seems impossible for the Axis to even compete in WWII unless the pact between Germany and Russia sticks, because otherwise, they are going to run out of oil before they have gotten further than Poland and eastern France.  I would also hope that they would remember their own history and take in proportionately tons of refugees.  If the Irish Palestinian state is weak during WWII, I would see them sadly getting conquered and treated very poorly.

The next interesting question is how long Ireland still recognizes them and their descendants as citizens and vice versa?  Constantinople and Jerusalem under Emperor Fitzgerald would look a lot more economically attractive than early 20th century Ireland!  On the other hand, if Irish Palestine fares poorly in the world wars, a lot of people would be interested in moving back to Ireland when their economy finally takes off in the late 20th century.