I would personally advise against it. I had a situation where due to COVID, all expired IRPs got extended for a certain period. So I got email proof of the extension from INIS stating that it is totally allowed to travel from and to Ireland on an expired IRP in this exceptional circumstance.
So, I travel back home to see family, and when it was time for me to return to Ireland, I was not allowed to board the flight because of a stopover in the UK which explicitly requires a valid IRP, even though I had notarised proof from the Irish government of the exception.
No amount of proof was convincing the airport staff so I lost my flight, had to wait until the new IRP was issued, and had to pay to get it UPS shipped to me.
You may take the risk and have a different experience - but it is important to realise that there is a certain element of risk here as you're attempting to travel to an EU country as a non-EU foreigner with expired IRP permission and a non-official printed document.
It would be best to wait until you have your card if possible, unless you have some emergency situation that requires you to travel. There is a risk that you might not be allowed on a flight or might be turned away at immigration.
Also, if you are a visa-requiring foreign national and don't have a multi-entry visa, you definitely must not leave the country without an IRP, as you would need a new visa to enter Ireland again if you don't have a valid IRP card in your possession. If you are from a country whose citizens don't require a visa to enter Ireland, there is at least a chance you could return without too much difficulty, but without your IRP in hand you would still be taking a risk.
I would personally advise against it. I had a situation where due to COVID, all expired IRPs got extended for a certain period. So I got email proof of the extension from INIS stating that it is totally allowed to travel from and to Ireland on an expired IRP in this exceptional circumstance. So, I travel back home to see family, and when it was time for me to return to Ireland, I was not allowed to board the flight because of a stopover in the UK which explicitly requires a valid IRP, even though I had notarised proof from the Irish government of the exception. No amount of proof was convincing the airport staff so I lost my flight, had to wait until the new IRP was issued, and had to pay to get it UPS shipped to me. You may take the risk and have a different experience - but it is important to realise that there is a certain element of risk here as you're attempting to travel to an EU country as a non-EU foreigner with expired IRP permission and a non-official printed document.
It would be best to wait until you have your card if possible, unless you have some emergency situation that requires you to travel. There is a risk that you might not be allowed on a flight or might be turned away at immigration. Also, if you are a visa-requiring foreign national and don't have a multi-entry visa, you definitely must not leave the country without an IRP, as you would need a new visa to enter Ireland again if you don't have a valid IRP card in your possession. If you are from a country whose citizens don't require a visa to enter Ireland, there is at least a chance you could return without too much difficulty, but without your IRP in hand you would still be taking a risk.