Abdulmecid was one of the more reform-minded Sultans, and they'd have a reason to be loyal to him personally. With time, it depends on the success of his reforms ITTL. Like the Hapsburgs, the Ottomans sometimes pursued a non-ethnic ideology--Ottomanism, in this case--that could keep the Irish loyal.
Given British meddling in the southern part of the Empire, and hostility from militantly anti-Catholic Russia, I could actually see the Irish forming a source of particularly loyal recruits to the Ottoman army--they'd have no particular inclination to support Serb or Romanian Orthodox rebellion, nor any attachment to Russia. They'd also have no reason to sympathize with Arab nationalists. If there are enough immigrants, I could see entire formations of the Ottoman army composed of Irish troops. Of course, that might not look good for the Sultan, to crack down on Muslim rebels with Catholic troops, so perhaps they might be limited to service in the Balkans. Irish troops bloodied in Ottoman service may well form the nucleus for Irish militant organizations in the late 19th/early 20th centuries.
Mind you, I don't think a mass movement of Irish to the Ottoman Empire is likely--urban jobs were also a draw for the Irish IOTL, and the Ottoman Empire's industrialization was anemic compared to that of the US.
EDIT: Indeed, if there is a WWI along roughly-OTL lines, you might find the strangest possible juxtaposition--the Ottoman Sultan-Caliph declaring Jihad against the Entente, while his Irish soldiers wage their own Holy Crusade against the anticlerical forces of Italy and France. Now that's ecumenical dialogue!
I was thinking there could be special irish fighting unites instead.
This also makes me wonder the effects on the homeland of Irish England, knowing their government tried to stop the donation and seeing Irish leaving to the Ottoman Empire, the English might lose favor with the Ottomans.