While as Yanniboy says, hospital records pre-1968 were destroyed, you can still obtain the Delivery Ward extract from
info@nmh.ie through an FOI request.
I did this for my step-grandfather who was born at Holles St hospital in 1929. It didn't reveal a lot of information, but does include info such as health of child on delivery (plus, weight, appearance of child, etc), parents details and who was present at the birth.
You will need the birth certificate & death certificate of the person whose delivery ward extract you're requesting, and a scan of your photo ID.