According to this GOOD Infographic, Irish travellers – along with those from Canada, France, Japan, Norway, Portugal and Spain – have no-visa access (or get a visa on arrival) to 170 countries. 
Those from Finland, Sweden and the UK do best (173 countries), followed by Denmark, Germany, Luxembourg and the USA (172), and Belgium, Italy and the Netherlands (171).
At the bottom of the table (no surprises here) are Pakistan and Somalia (32), Iraq (31), and Afghanistan (28).
