The Irish went everywhere including the Ottoman Empire. The modern nation of Turkey is much smaller than its predecessor empire, which was dismantled after World War I.
From 1583-1825, the Levent Company Merchants from England held by charter a monopoly of British trade with the Ottoman Empire. They had major trading posts at “Scanderoon” (Aleexandretta), Smyrna and Aleppo (now in Syria). The Irish would follow in their footsteps.
The Sultans in the nineteenth century began to modernize the empire. As a result, Europeans were invited to help, with Symrna (Izmir) and Constantinople (Istanbul) becoming gathering places. The collapse of the Ottoman Empire, and the Turkish Revolution brought an end to this era.
When foreigners intermarried with the local Christians, their children were legally considered Turks, and not foreigners. The French coined the term Mezza Razza to describe these mixed families. The Irish were certainly among the Mezza Razza.
The Ottoman Empire was a mixture of cultures and peoples. Christian communities included Orthodox (Greek, Armenian, Russian), Catholics (Roman, Armenian, Greek/Eastern Rite), and most Protestants attended the Anglican Church. Large collections from all these denominations are on microfilm at the Family History Library
(FHL): www.familysearch.org Make no mistake, the Irish are recorded in them. Also don’t neglect the British Consulate records for vital records; also at the FHL.
Noteworthy collections are deposited in London. These include the “Levant Company’s Archive (1634-1825)” at the Guildhall Library: www.cityoflondon.gov.uk and the “International Memoranda (1821-1890)” which are Church of England records housed at the National Archives, Kew: www.nationalarchives.gov.uk
Just be aware there are reasons why an Irish family would be in the Ottoman Empire. Also be aware that your Middle Eastern Christian family may indeed have an Irish connection. There are historical currents to explain all this with records to back it up.