Ireland is part of an island. By definition islands are not part of any continent. Long Island is not part of North America, although it part of the United States of America. So is Hawaii, but that's not not part of North America.
Actually islands that lie on the continental shelf are usually considered part of the continent in which self they are on. Thus Long Island is part of North America as well as Ireland, since it is on the European continental shelf. Also geographers continually add Ireland to the continent of Europe since it is on the European sub-plate, which is itself part of the larger Eurasian plate.
The only point in the statement above that has validity is Hawaii which is not part of the continent of North America since it is on the Pacific plate. It is usually labeled as part of Oceania or just the Pacific.