Some 300 staff at the banks information technology division began a 24-hour strike after prolonged negotiations on the terms of their transfer to the US IT giant broke down.
The Irish Bank Officials' Association (IBOA) has also called an overtime ban until the dispute is resolved.
Job security for those who transfer to HP is at the heart of the dispute. Bank union leader Larry Broderick said, "If Bank of Ireland and Hewlett-Packard will get benefits over seven years from this contract, why can't the staff be given the option of redeployment to the bank over that period?"
The union is calling for alternative work in the bank for staff who do not want to transfer to HP and the right for any staff who are unhappy with the outsourcer to return to the bank during the life of the contract. Bank of Ireland said the union's demands were unreasonable.
Bank of Ireland and HP have guaranteed no compulsory redundancies for two years from the date of transfer and promised a further year's salary in the event of compulsory redundancy.
IT staff have also been offered a minimum of €3,500 (£2,450) in compensation for their transfer.