Your facts are not clear my friend with regard to dual citizenship. You can only be a citizen of one country and not two as many people feel that they can be citizen of two countries. So you can be a Green card holder of USA but if you give up your citizenship of India and take USA or Ireland citizenship then you can apply for OCI card for india.

Yes, you will cease to be an Indian citizen immediately after you obtain citizenship of any other country including USA or Ireland.