Some asylum-seekers have had to sleep on the streets of Dublin because officials could not provide them with accommodation.
The crisis has come as refugee numbers climb steadily. More refugees have arrived in the first nine months of this year than in the whole of 1998.
In one instance a couple with three young children slept in a park when the refugee application centre turned them away. They appealed to the Dublin office of the United Nations High Commissioner for Refugees, and were put up in a city-centre hotel.
Last night a UNHCR spokesman expressed concern for asylum-seekers.
Sources say Eastern Health Board staff in the refugee applications centre in Lower Mount Street, Dublin, started turning away single people last week and have since extended this to families. The centre is run by the Department of Justice and houses Eastern Health Board and medical staff.
The EHB acts as agent for refugee accommodation but is in fact responsible for providing it. The Department of Social, Community and Family Affairs foots the bill.
Asylum-seekers are arriving in Ireland at a record rate of 250 per week. Numbers have doubled since last July.
Meanwhile, the Taoiseach joined other EU leaders at a meeting on justice and home affairs in Tampere, Finland. The UN High Commissioner for Refugees, Ms Sadako Ogata, said that the fight against illegal immigration must not become an excuse to diminish the rights of asylum-seekers fleeing persecution. In 1998 fewer than 30,000 people were granted asylum across the 15-member EU.