Bank of Ireland staff in 100 branches will no longer handle cash by the end of the year.
Customers will still be able to lodge and withdraw money from self-service machines, but will not able to take out more than €1,300 or deal with foreign currency.
The bank said at the moment just 3% of transactions are conducted over the counter.
Justin Moran from Age Action Ireland said the move is disappointing, particularly for older people. He pointed out that the majority of people aged 65 and over have never been online and do not know how to access online banking.
"Many of them [aged 65 and over] want to do their financial business over the counter, they want to do it with bank staff they know and trust with whom they have relationships with.
"We'd been worried that thousands of older people are going to be left behind as Bank of Ireland leads and other banks follow trying to push people to doing as much as their business online as possible."
Larry Broderick from the Financial Service Union says they have two main concerns.
"For the staff, are they adequately staffed, are the proper resources in place," he said.
"And obviously for the customer, we're very conscious of the aged, we're very conscious of the customers that have cash, and we're engaging with the bank to try and see how we can factor those interests as part of a new plan."
He added: "There's a number of models out there that look at what the future of banking would be.
"But the fact of the matter is that this is about providing a service, and if you want to be a community bank, you have to factor in all those people in the community."