You're in a fix for sure. I hope you've discussed this with him. I do a lot of family law, and if your boyfriend is this determined to keep you out of the home's finances then that's warning bells for me. It may not just be pride; it may be that since you're not married and he pays the mortgage and taxes, he's setting up a situation whereby you have no claim to any equity in the home if you separate.
I'm not suggesting anything about your relationship. But a big part of my job as a lawyer I say is "to spot your worst case scenarios which you haven't considered, and tell you how to avoid them".
I'd hate to think that you're going to throw down thousands of dollars here, when it's clear that his income can't keep things going, when he doesn't want you to, and have that lead to more tension and a breakup. After you've spent all your money on "his" house.
So if you haven't had a discussion with him about this first, I think you should before you pay anything.
The bank won't question you about why your boyfriend hasn't paid. You're not responsible to pay, so they don't want to hear it from you. And they don't really care anyway why he's behind, they just want their money.
Now, how badly they want their money is going to determine how much negotiating and information sharing they will do. Since you're not on title or on the mortgage, the bank shouldn't be talking to you about this at all. So if they say "we'll gladly take your money and apply it to the mortgage, but we can't discuss any details because you're not a party to the foreclosure or the mortgage contract" you'll know why. However, if they want their money they may well talk to you about. Ask them what's the minimum to forestall this foreclosure proceeding. They'll say they need it all. Ask how much that is. When they tell you, offer half (if you can pay half). Don't offer to go on the mortgage or commit to further payments until you've discussed this with your boyfriend.
Three years behind on property taxes will result in lien on the property, it's probably already on title. The municipality has to be owed a lot before they force sale of a home for municipal taxes. If and when the mortgage is brought into good standing, your boyfriend could access the equity in the home to pay the taxes. However, the bank is unlikely to give him a good rate on a Line of Credit given his recent payment history, and to renegotiate the current mortgage before it's up for renewal, or get a second mortgage, is going to incur fees.
The bank might well relent if you volunteer to go onto the mortgage too. That's extra security for them. Which is what they want under the circumstances. However, I beg you to think long and hard about this. Being on the mortgage doesn't put you on title to the home, and since you're not married you have very few automatic rights to possession of or equity in the home (without going to court, anyway).
I'm trying to answer your questions but clearly I'm hearing warning bells all over the place about your situation. If your boyfriend makes good money, you have to wonder why he's not up on the payments; is there a substance abuse or gambling problem, or paying support for children you don't know about, or has another girlfriend on the side he's paying for, I don't know. But please, sit down with him and talk about this, ask the hard questions, say that if you are going to help now and in the future you need some assurances that he's spending his own income wisely because now it's affecting you too.
Does that make sense?