Principals are struggling with unusually high absences of teachers and special needs assistants (SNAs), who have either contracted the virus or are a close contact of a confirmed case.
Up to last night and early today, many were still frantically trying to find substitute cover to allow all classes go ahead as normal.
But education sources were predicting that some schools would not be in a position to bring in all classes because of a lack of staff.
Schools will face similar problems in the days and weeks ahead, leading to significant disruption to teaching and learning.
About 4,000 third- and fourth-year student teachers can now work as subs, as part of a range of emergency measures to help fill gaps, and there was brisk activity on the Teaching Council website yesterday as they signed up to the professional register.
Children with special needs, younger primary pupils, as well as third-year, fifth-year and sixth-year classes at post primary will be prioritised for in-person teaching if schools have to ask classes to stay at home.
Where classes are not in-person, schools have been told to make every effort to provide remote learning for pupils.
Teachers doing online classes will be expected to engage with pupils every day.
Department of Education guidance issued yesterday spelled out the various channels open to principals seeking substitutes, and, if none can be found, they will have the discretion to ask classes or pupils to stay at home.
This flexibility relates only to staff shortages, and the usual arrangements will apply to situations where a Covid case is confirmed in a school.
As many as 8,000 primary teachers - about 15pc - will not be available for work today, according to Irish National Teachers Organisation (INTO) general secretary John Boyle.
John Curtis, general secretary of the Joint Managerial Body (JMB) representing about half of second-level schools, said: "We know schools will do everything in their power to reopen for all of their students, but we know of some difficulties."
Páiric Clerkin, CEO of the Irish Primary Principals Network (IPPN), said, "There will be difficulties."
Of 1,500 schools that responded to an IPPN survey on Tuesday, 40pc were concerned about the level of staff shortages.
Paul Crone, Director of the National Association of Principals and Deputy Principals (NAPD), said the situation "is on a knife-edge. Schools are opening, which is the right thing to do, and principals are making decisions as best they can".
One of the dilemmas facing principals is at what point to accept that a substitute cannot be found and to tell parents that a class cannot come in to school.
Mr Crone said decisions like keeping certain classes at home would have to be made the night before.
"If students leave their house in the morning and arrive at school, you can't turn them around," he said.
Michael Byrne, principal of Citywest and Saggart Community National School, Dublin, feels principals in some instances can't make decisions on closing classes the night before. He said they often don't know about staff absences until the morning.
Speaking yesterday, he said: "Our position now is we have 15 members of staff out tomorrow. We have been able to cover everything but just about.
"I met my deputy in the school today and at the time we arrived there were eight members of staff out, and by the time we left, there was 15. As the day goes on, we're just waiting for more.
"At the moment, every class can come in, but no child with additional needs will be given their hours."
He said they were lucky, but smaller schools are having to prioritise and close classes.
Aaron Wolfe, the principal of Coláiste Éamann Rís, a secondary school in Cork city, said: "It is proving very difficult to keep the school open. I have 13 staff members out tomorrow.
"I'm fortunate we have a huge staff. I have 480 students, 50 teachers and 20 SNAs."