Irony is when

1. My bank wants me to visit its branch to update my email. I wonder why I can't convey my new email address to my bank via Internet Banking or in a letter signed by me in 'wet ink'.  
2. Left with no choice, I visit the branch but find that it has moved.
3. Despite 'friending' it on Facebook, I keep getting irrelevant offers from my bank on my newsfeed, but when its branch relocates, it doesn't notify me of such an important event in advance. When I ask my bank if payors who customarily make electronic fund transfers to my account need to update the branch address field in their 'List of Beneficiaries' screen, my bank is clueless.
4. EFTs can happen in one hour but my bank won't let me initiate one for 24 hours. The National Payments Corporation of India (NPCI) has established a net settlement system that can complete a retail payment transaction within one hour by running over 10 settlement cycles during an 8 hour window. However, my bank takes 24 hours to approve my request to add a new payee, without which I can't initiate the ePayment. It doesn't bother to explain how this 24 hour 'cooling period' benefits the payor, payee or itself. Since my account doesn't get debited until the next day, float - the usual suspect in such cases - can't be responsible for the hold up.
5. ePayments need cheques. Apart from providing my bank account details, I'm required to submit a canceled cheque before a payor can transfer money to my account electronically. Asking payors why this step is required is pointless since all of them uniformly blame it on 'security measures imposed by the regulator'. It's quite dumb if the payor requires the canceled cheque to establish that I'm the legitimate owner of the said account - I'm not stupid enough to falsify the bank account details and see my money getting transferred to someone else.

I'd love to hear more examples of ironies in Internet, mobile and social media banking, whether nationwide or with a few specific banks.