People are being urged to avoid calling the Digital Covid Certificate helpline unless their queries are urgent or they are travelling in the next ten days.
Fewer than 3,000 out of 40,000 calls to the helpline over the past two days were dealt with.
Taoiseach Micheal Martin said the helpline had become "overwhelmed" this week.
He said that people who were travelling in September did not need to call the centre, adding there had been a variety of calls "some necessary, some not so necessary".
Speaking to reporters after today's Cabinet meeting, he said that the issuing of 2.1 million Digital Covid Certificates was a "significant achievement".
Liz Canavan, a senior official at the Department of the Taoiseach, said there are "long waiting times for many callers and the centre has not been able to meet the level of demand".
Addressing the Oireachtas Committee on Health, Ms Canavan said many people who are calling are not due to travel in the short term and "these non-urgent calls are adding very significant waiting times to all calls, and preventing many urgent calls from getting through".
Ms Canavan said the helpline is also getting calls from people with requests for certificates for PCR or antigen tests, which, she said are generated by third party providers and not the Government.
"We are urging people to only call the call centre if they have an urgent call and are travelling in the next ten days. It can only support queries in relation to vaccination certificates and recovered certificates."
The committee heard there are currently 55 agents answering calls. This will be increased to 85 tomorrow and to about 100 this day next week.
Ms Canavan said more than 40,000 people called the number on Monday and yesterday, including 34,000 yesterday alone. However, just 2,826 queries were handled.
She said the high volume of calls "far exceeds estimates of call centre demands, based on estimates of travel bookings and survey or insights data on travel intentions that the Government and industry had access to".
She said the rapid issuing of 2.5 million certs in just over a week has also contributed to the demand.
Sinn Féin TD David Cullinane said that to argue that officials could not have anticipated the volume of calls "is just to stretch people's patience".
Meanwhile, the Irish Travel Agents Association (ITAA) warned that although authorities in Malta are accepting EU Digital Covid Certificates, they do not recognise HSE vaccination cards, which has led to a number of Irish tourists facing quarantine on arrival despite being fully vaccinated.
It has urged the Government to issue Digital Covid Certificates to all those affected.
The ITAA has advised prospective holidaymakers to check the rules and regulations of their destination country before departure, and also to check in with their travel agent for the most up-to-date advice in international travel.
ITAA President Paul Hackett said: "To the best of our knowledge, Malta is the only EU country that is refusing to accept HSE vaccine certificates.
"This is a situation that can be easily and quickly rectified, if Government issue DCC's to all affected travellers. The DCC helpline is also under-resourced, which must be addressed as international travel continues to reopen."