Ireland is absolutely part of the British Isles, just not part of Great Britain. I would say that it's generally only considered correct to call someone from Great Britain British, rather than the Isles as a whole though. However, in common parlance I would say that people from Scotland and Wales use Scottish and Welsh more than British, with people from England using English and British interchangeably, and people from Northern Ireland (that are unionist anyway) using the term British over Irish. That's all to say, you'd probably get a smack upside the head for calling someone Irish British, and rightfully so.