The total daily withdrawal limit for your debit card is set by your bank. It varies from bank to bank and even account type to account type at the same bank. You have to ask them about your specific accounts, but I think you will be surprised at what the account limit actually is. Many banks are flexible enough to set your withdrawal limit higher, temporarily like for your trip or permanently, if you ask.
There is a totally separate limit to think about and that is the limit imposed by the ATM itself which is where the $300 limit comes in in the states. It just happens to be the most $20 bills an ATM can dispense in one bunch. You might have a $1000 limit on your card, but the specific ATM you use in Ireland might limit individual transactions to €200. If you run into this, and you know you have not exceeded your daily account limit, just try another transaction for €200 and keep going if you need more. You can also go to another ATM if one quits giving you money.
Make sure you inform your bank of when you are going and what dates you will be there. Otherwise they will probably deactivate your card suspecting fraud.
Oh, and avoid using your debit card to do anything other than get cash from an ATM except in emergency situations. It's just simpler that way.