British Newspaper Archive database. They are:
Saunder's News Letter, a paper published from 1755 to 1879. Some 1,838 editions have been uploaded this week. The majority date from 1783 to 1787 when the paper was published six days a week. Less frequent editions are also available for February to November 1794 and January to December 1795.
Cork Advertising Gazette. All 206 editions from October 1855 to October 1859 have been uploaded for this weekly paper. I'm not sure if the paper stopped publishing or if these are the only surviving editions, but it will be the full extent of the holding for this title.
By my reckoning, there are now 70 Irish titles in the British Newspaper Archive. They're all available as part of FindMyPast's Ireland and World subscription packages, too.