My feeling is this: if you have distant relatives who may have been from Ireland, lived in Ireland, etc, and you would LIKE to say you're part Irish....then you're part Irish. Hell, if you go back far enough, we're all bloody related, so if you want to be Irish, what the hell; go for it.
This is how I feel. Countries and tribes are social constructs. Families are genetically linked. Genealogy is interesting because it's about individuals and history. My grandfather was born in Ireland, and his story is interesting to me, but I was born in Boston. I also have recent ancestors from England and Norway. I feel no special connection to the cultures of Ireland, England or Norway-- my culture is the United States of the Groovy 60s-- but I can claim Irish, English and Norwegian ancestry.