If a county, tax collector's office, puts a lien on a rental home for failure to pay property taxes and sewer bill. What can happen to me, the tenant? Should I start looking for a new place to live?

No, most likely it will be a couple of years before the home would go up for sale over taxes. And then if there is a mortgage on the home, normally the lien holder will pay the taxes long before they are sold for taxes.
** you can call and ask the tax department their procedure on this.