That's not true in general. People in Northern Ireland are entitled to one or both citizenships which makes it true in that limited case.
http://www.electoralcommission.org.uk/faq/voting-and-registr...
"To vote in a UK general election a person must be registered to vote and also [...] be a British citizen, a qualifying Commonwealth citizen or a citizen of the Republic of Ireland"
http://www.dublincity.ie/yourcouncil/votingandelections/Page...
"Elections for Dáil Éireann: Both Irish and British citizens can vote in elections for Dáil Éireann (the lower house of parliament)."
That's not true in general. People in Northern Ireland are entitled to one or both citizenships which makes it true in that limited case.