You will need to consult with the consulate of each country you are interested in visiting. Canada, for example, specifically asks on its entry forms if visiting persons have any criminal conviction, and goes on to specify that even minor offenses, such as DUIs, must be disclosed.
Failure to accurately disclose the conviction prior to entering a foreign country may be a crime under that country's laws. The only accurate way to determine the answer is to ask the consulate of the country you intend to visit. The Republic of Ireland's consulate has a link with phone numbers for their offices: http://www.discoverireland.com/us/ireland-plan-your-visit/facts/Visa-Passport-and-Embassies/