Good luck with this. It is a difficult situation.
I've never heard of a county being able to go after a tenant for taxes, but laws change all the time, which is why I suggest you consult an attorney. Also, ask the lady you've been talking to for the exact statue she is referring too, and what the penalties are if you don't pay the taxes.
Once again, I'd strongly suggest you speak with a real estate attorney to protect your interests. Good luck.
I would strongly recommend you take a copy of your lease and the letter from the tax office and talk with a real estate attorney to make sure you are doing what you are supposed to do. It may cost you a little, but protecting your rights and credit is worth the investment.