This was previously the case. Article 44, Section 2 of the 1937 constitution stated

“The State recognises the special position of the Holy Catholic Apostolic and Roman Church as the guardian of the Faith professed by the great majority of the citizens.”

This reference was removed in 1973 by means of a referendum. Article 44, 2.2 states that “The State guarantees not to endow any religion.” However, the influence of the Catholic Church in Ireland seems to run counter to this article. For example, the preamble to the Irish Constitution is as follows:

“In the Name of the Most Holy Trinity, from Whom is all authority and to Whom, as our final end, all actions both of men and States must be referred, We, the people of Éire, Humbly acknowledging all our obligations to our Divine Lord, Jesus Christ, Who sustained our fathers through centuries of trial…”

Further to this, the President of Ireland must make the following declaration before entering office:

“In the presence of Almighty God I ,do solemnly and sincerely promise and declare that I will maintain the Constitution of Ireland and uphold its laws, that I will fulfil my duties faithfully and conscientiously in accordance with the Constitution and the law, and that I will dedicate my abilities to the service and welfare of the people of Ireland. May God direct and sustain me.”

Judges and members of the Council of State must take a similar oath. The oaths appear in Articles 12, 31 and 34 of the constitution.
Aside from the constitution, the Church’s influence manifests itself in a number of other ways. For example, the ‘Angelus’ is observed every day on RTE radio and television, the state’s broadcasting service. The sale of alcohol is also prohibited on Good Friday.

The Church continues to exert a huge influence on the Irish Education system.
Any sufficiently large group of people in the state can apply to open a school in their local area; the Department of Education chooses amongst such applications based on need. Up until 2006, groups wishing to setup a school needed to contribute 15% of the cost of the building.
So for both financial and historical reasons, the majority of Irish primary and secondary schools have been managed by the Catholic Church. This situation continues today, such that Catholic bishops or their designated representative act as the patron of 93% of all primary schools. Traditionally, many teachers in these schools would have been priests, nuns and brothers. However, declining vocations has led to a situation where most staff in Catholic schools are in fact professional teachers.
Despite this, the Church still wields significant power in the management of schools. A school is typically run by a Board of Management, setup at the discretion of the patron. The Board of a large school is usually composed of 8 members; 2 direct nominees of the patron, 2 elected by the parents of the pupils, 1 member elected by the teaching staff, the principal of the school and two additional members agreed by representatives of the patron, teachers and parents. The two community members must make a committment to uphold the ethos of the school. The patron then appoints the chairperson of the board, which is typically the local parish priest in the case of a Catholic school.
The two main powers of the Church in education are in enrolment and recruitment. The Equal Status Act (2000 – 2004) outlines 10 grounds on which discrimination is outlawed, but gives church-controlled schools the right to discriminate in two of these areas: gender (so a boy cannot enrol in a girl’s school) and religion. It is perfectly legal for a church-run school to refuse the enrolment of an unbaptised child in favour of a Catholic one. Often the Church-run school is the only one available for parents and anecdotal evidence suggests that some parents have had their child baptised in order to secure enrolment (referenced, for example, in this Equality Authority position paper).
A church-run school also has the power to fire a member of staff if they decide that they are not upholding the ethos of the school. Technically a teacher could be fired for having a child out of wedlock, being gay, etc. Although it would be unlikely to happen today, there is a recent historical precedent in the case of Eileen Flynn. A report by Marguerite Bolger suggests that firing someone on these grounds remains legal. Moreover, we’ve been sent messages by users and found many posts on the “Education Posts” forum in which teachers express concern about revealing aspects of their personal life – pregnancy outside of wedlock, for example – to the school authorities or board of management. Regardless of the reality of these fears, the fact that the question is being posed is in itself interesting.
We spoke to the principal of a primary school who told us that, although it is illegal in Ireland to ask someone in an interview what their religion is, that the issue is often raised in an indirect fashion. Referring to interviews he had participated in earlier in his career, he told us that candidates are often asked if they have gone through an exam in the Catholic Catechisms and that

“The only appropriate responses to this question are “Yes” or “No, I haven’t yet, but I am a practicing Catholic so I wouldn’t have any trouble anyway…”.

Moreover, he told us that, in his experience, teachers increasingly include a reference from their Parish priest in their job application and that of 200 applications for a position he advertised, 75% of applicants included such a reference.
Aside from these two areas, the Church’s influence extends, unsurprisingly, into the domain of religious education. All schools in Ireland must devote at least 30 minutes a day to the teaching of religion. Pupils who are not Catholic must either sit out these minutes separate from the rest of the students, or listen without participating. One could argue that this situation has a relatively modest effect on the teaching of non-Catholic pupils since, if they listen in, basic religious instruction would likely focus on issues of right and wrong that are applicable across the board. However this changes in the run-up to Communion and Confirmation, when a large amount of time is set aside for preparation and the relatively “ethics based” coursework of the Alive O curriculum is replaced with preparation for receiving the sacrements. During this period, non-Catholic children are obliged to miss large amounts of class time. Moreover they are separated from their classmates and marked out as different from a very early age.
Finally, it is worth noting that all taxpayers, regardless of their religion or beliefs, pay taxes that fund our schools. Although church-run schools receive no more money than any other, the fact is that non-religious tax-payers are indirectly paying to fund a system where a large proportion of school time is spent preparing for religious rituals that they may not believe in or that they do no wish their child to take part in. Were there a choice between a state secular school and a Catholic school this would not be a problem. But with 93% of primary schools church-run, there is no choice.
Note that Educate Together multidemominational schools are now the fastest growing schools in the state.

Count Me Out have written to the Department of Education and Science requesting these figures and are awaiting a response.

Catholic bishops are patrons of 93% of primary schools in Ireland. In general, the patron of a school is a representative of the owners and can be an individual or a group.
The Education Act 1998 gives a statutory basis to the role of the patron and sets out the rules for determining who the patron is. The patron may manage the school personally or may appoint a board of management to act as manager. Under the Act the patron has the power to remove the board and take over managing the school or appoint another board. A register of patrons is kept by the Department of Education and Science so it is possible for anyone to check exactly who the patron of any national school is.

They must obey the act, but only in so far as it does not conflict with their religious ethos.

The Ryan Report was published in May 2009, and there is a good history of its origins and operation on Wikipedia.
You can download a copy of the report from the Child Abuse Commission website.
Moreover, there are a number of good articles in The Irish Times. Namely:

• “The 18 Religious Orders – What they do now”;
• “Lessons in the Power of the Church”;
• “Moved to outrage by apathy over child abuse”.

The report on the Dublin Archdiocese was published in November 2009, and you can read more about its history on Wikipedia.
You can download a copy of the Report from the website of Department of Justice, Equality and Law Reform.

The largest number of multi-denominational primary schools in Ireland are run by Educate Together. There is a full list of their schools here. Count Me Out have written to the Department of Education and Science in order to get a list of all the multi-denominational primary and secondary schools in the state.

Religious orders still own and operate many of the hospitals in Ireland.
For example, in Dublin, the Sisters of Mercy operate the Temple St. and The Mater hospitals, while the Sisters of Charity operate St Vincents and St Michaels. All of these hospitals must uphold the ‘Catholic ethos’. There are a number of instances where modern medicine can come into conflict with this ethos, such that the operation of the hospital has been disrupted. For example, in 2005, The Mater hospital’s clinical advisory group refused to allow the trial of a lung cancer drug to proceed because part of the treatment would have required patients to use contraception. You can read more about that incident here and here.
There are older examples of the Church’s ethos clashing with medical practice. For example, between the 1940s and 1980s symphysiotomies were performed on pregnant women in preference to caesarean sections because they were deemed to be more in keeping with the Catholic ethos. More information on this can be found here and here.

In schools, the patron wields a significant amount of power. For example, a school’s board of management requires the approval of the patron for the appointment of teachers. As stated above, 93% of primary schools are under the patronage of the Catholic bishops. Patrons are entitled to veto decisions which they feel are not in accordance with the Catholic faith.
Some high profile examples of the Church’s use of power in the provision of state services are:

• In May 2009, a Christian Brother in a Dublin secondary school requested that government-issued anti-homophobic bullying posters be removed from the school premises (ref: Senator David Norris senate speech, May 27th 2009). The group involved in creating the posters – BeLonGTo – have confirmed to us that the incident did occur as recounted by Senator Norris;
• In 2005, the ethics committee of the Mater hospital, which is run by the Sisters of Mercy, refused to approve clinical trials of Tarceva (a lung cancer drug) because the trial required women to avail of contraception (text of the article can currently be found here). This is against the hospital’s ethos;
• A principal in Meath, Tomás Ó Dulaing, was sacked in 2002 by the board of management for seeking to have religious education outside of class time;
• A teacher in Wexford, Eileen Flynn, was sacked by the board of management from her post in 1982 for having a child out of wedlock.

The most effective way to reduce the Catholic Church’s power and influence in Ireland is to firstly decrease their membership. Defecting is primarily a symbolic act which will allow you to leave the Church for your own personal reasons. Obviously, reducing the membership will not automatically reduce the Church’s power. The most important indicator of religious affiliation in Ireland is the census, which is conducted every 5 years.
The interest that the Count Me Out campaign has generated has resulted in a debate on the whole relationship between church and state. This debate has allowed a lot of people an opportunity to reflect on their relationship with the Church. As such, more people may be willing to be open about the fact that they wish to see a diminished role for the Church (or religion in general) in state affairs. One of the best ways of realising this is for the state to see that the number of Catholics in the country is decreasing. Marking ‘No Religion’ on the census will achieve this. Incidentally, according to the census figures, ‘No Religion’ is the fastest growing category under ‘religious affiliation’ in Ireland since 1961.