SIPTU is urging the national ambulance service and management of the Health Service Executive not to implement the new Eircode system.
The union says the postcode system should not be used until it is proven to be fit for purpose.
Freight companies including FedEx and DHL say they will not use the service because it is 'outdated' and the unique seven-digit codes are not searchable on Sat Navs or Google Maps.
While there has also been complaints that some homes are being marked in the wrong towns, counties or provinces - including locating Shannon Airport in Co Limerick, instead of Co Clare.
The new service is optional and will not replace any existing lines of addresses.
Eircode says 2.2 million individual numbers will be posted to properties in the coming days.
But Paul Bell, health division organiser for SIPTU, told Newstalk Lunchtime the glitches should be worked out first.