The bank is making the payments after its internal monitoring processes flagged that ATM users who forgot to take their cash from the machine were still having the money taken from their accounts.
When distracted customers fail to take cash or their cards at ATMs, the machine waits a short while before pulling the money back in with the debited amount automatically credited again.
However, for four years, until October 2009, the installation of anti-fraud measures "resulted in the normal system response of automatically prompting a refund to issue not to occur for these particular transactions" says Bank of Ireland in a statement.
It is now in the process of refunding €1.3 million to 14,000 of its own customers this week and is working with other banks to help get a further €1.7 million to 29,000 non-customers.
The bank says has apologised for the error and has informed the country's Financial Regulator.
About six people per ATM per year leave their cash, card, or both in machines.