I havent noticed any requirement to have an Irish passport. The requirements mentioned in the booklet you can download are:
(i) Be a national of a European Union Member State; or
(ii) Be a national of a European Economic Area State or the Swiss Confederation; or
(iii) Be a Refugee under the Refugee Act, 1996; or
(iv) Have had a period of one year‟s continuous residence in the State on the closing date of
the advertisement for the competition for the vacancy to which the admission relates, and
during the eight years immediately preceding that period, has had a total residence in the
State amounting to four years; or
(v) Has been granted subsidiary protection, or is a family member of such a person, in
compliance with the Admissions and Appointments Regulations 2013.
http://www.publicjobs.ie/publicjobs/cam ... ooklet.htm