I regularly transfer money from my BoI current account to my BoI savings account. Firstly, you will need a savings account and need both your savings account and your current account registered with Banking 365 and the one you want to transfer to registered as a beneficiary as boaber has mentioned. (). Then just transfer to the savings account like you would to any other beneficiary like in the . Bear in mind though, you mightn't be getting the best rates on a regular savings account. You might be better to set up a Special Bonus Saver Account and modify your direct debit monthly. You can still see your balance online but can't do the transfer online (last time I discussed it with them anyway).
Bank of Ireland aside, there's nothing to stop you registering any type of savings account in any bank and transferring the money using BoI's Banking 365 assuming that's where you want to transfer the money from.