He added: "When I go to a Roman Catholic church I respect that tradition - that we do not go forward for communion.
It comes only days before Archbishop Desmond Connell's elevation to Cardinal by the Pope.
Dr Connell was first involved in controversy over the issue three years ago, when Irish President Mary McAleese took communion at a Church of Ireland cathedral.
Then he said Anglican ministers who welcomed everyone to communion were failing to respect the faith and obligations of Catholics.
His latest comments, in the Sunday Business Post newspaper have drawn a reponse from the leader of the Church of Ireland, Archbishop Robin Eames.
He said on Monday: "Holy communion is the deepest and most personal expression of worship for any Christian.
It is tragic that we have to view this service as a source of division between churches... surely we must strive to respect each other's deeply held convictions."
Dr Connell's remarks drew stronger criticism from Dublin's Church of Ireland Archbishop, Dr Walton Empey.
Dr Empey said he was saddened Dr Connell's elevation on Wednesday, which could have been a joyful one for the Catholic Church, was being marred by "acrimony".
He added: "When I go to a Roman Catholic church I respect that tradition -
that we do not go forward for communion.
"But when we are in our own Church of Ireland churches, then I have to ask them to respect our position on that matter."
Journalist and historian John Cooney said the reaction in Dublin was absolutely unprecedented in terms of interchurch relations.
He said: "Quite frankly the members of the Church of Ireland are furious over the remarks of Dr Connell."
Break with tradition
Archbishop Connell's appointment in January broke the Vatican tradition of conferring the honour of cardinal on the Archbishop of Armagh - the leader of the Catholic Church in Ireland.
Archbishop Sean Brady has held that position since the retirement of his predecessor Cardinal Cahal Daly.
Religious commentators say the appointment is a very unusual step which will raise questions about where power lies in the Church in Ireland.
Dr Connell has already flown to Rome for the ceremony to install him as a cardinal.
It is to be held on 21 February at the Vatican.