The school and college shutdown is being extended until Monday April 20 at the earliest, but no decision has been taken in relation to the State written exams in June.
The Leaving Cert and Junior Cycle exams are still scheduled to go ahead as normal, but that could change, depending on the spread of Covid-19 in the community.
Taoiseach Leo Varadkar announced the extension as part of a wider package of measures to restrict the spread of the virus.
However, Maynooth University (MU) will not be re-opening for face-to-face lectures for the rest of the academic year, it has announced.
The university made the announcement amid the uncertainty over how long the Covid-19 crisis will last.
Maynooth said the uncertainty has caused additional stress for some students.
Earlier, Education Minister Joe McHugh said they wanted to make the State exams to happen but they would be influenced public health officials.
Speaking on Galway Bay FM today, the minister said: "We don’t know where we are going to be in terms of the peak of this virus."
But he added: "I think we owe it to those young people who are studying and putting in the effort to see this through to try to have these exams this year. When I say this year, I mean the dates that are defined for this year. That’s what we are working towards."
In the worst case scenario, schools would not return until the end of August/ September, more than five months after closing their doors on March 12.
The extended shutdown leaves little time for post-primary schools, which wrap up by the end of May anyway, to resume in a meaningful way in the current academic year.
The primary school year continues to the end of June, so that offers more latitude for a resumption of classes at the country’s 3,000 primary schools, if circumstances allow.
While the orals and practicals, which were due to start his week, have been cancelled, running the written exams is now the top priority of the Department of Education.
However, if it is too problematic to run all the exams, the Leaving Cert will get priority.
Some 126,000 students are entered for the June exams – 61,053 due to sit the Leaving Cert and 65,190 at Junior Cycle level.
Institutes of technology ( IoTs) have ruled out the use of traditional exam halls for end of year assessments.
The Technological Higher Education Association (THEA), the umbrella organisation representing the institutes and TU Dublin, said assessments will respect social distancing.
Individual colleges will be in touch with their students about the revised arrangements for different courses, which will include online assessment and more extensive use of continuous assessment.
THEA said colleges were committed to ensuring that the cohort of 2020 was not delayed or disadvantaged through the current crisis.