This bill really seems excessive. It won't do him any good to bury his head in the sand, as other posters have already stated. However, I think he's entitled to a copy of his account from the ESB. I had a tenant in an apartment for six months and her bills were very high. She just paid them and didn't mention it to me until the day she was leaving. The same was true of her gas bills. When I looked at them, I realised immediately that both companies were sending her estimated bills! I got the management company to take meter readings and she was due hundreds back!
When your boyfriend left the apartment, his landlord should have given him the final meter readings. I would insist that ESB provide me with a copy of my account, especially if I had never received a bill. He should check that the balance on the account was zero when he took it over and that these bills were not just estimates. The ESB AFAIK are usually good about making payments when customers make an effort to pay. From their perspective, your boyfriend has ignored them for years, so they might be understandably tetchy with him initially!! He should definitely not ignore this situation as it really has got potentially dire consequences.