Legally it depends on the particulars of your warranty coverage, which varies by vendor. For instance iFixit offers a one year warranty on batteries, which seems to be fairly standard but it's vendor specific; some may be shorter or longer.
So if your warranty has elapsed, then generally you're out of luck. However, it would be worthwhile to contact the vendor and explain the situation to them and see if they would be willing to do anything for you. To be honest, at the best case you may be able to convince them to replace the defective battery, but I'm skeptical you'll be able to talk them into paying for repairs to whatever got damaged. I can't hurt to ask though; the worst that can happen is they say no and you're no better or worse off than you are now.