Generally, no. Depending upon how the business is organized, you will likely inherit one of the following:
Limited Liability Company Membership Interest
Sole Proprietorship Assets
With a corporation or LLC, what you really are inheriting is the net worth of the business. With a sole proprietorship, you inherit both the business and its assets.
For example, if the business is a corporation and you inherit the stock, the business still has all of its assets and still owes all of its debts. If the business were liquidated, the debts would have to be paid before you as a stockholder would receive a distribution. The same formula would apply to a limited liability company.
A sole proprietorship is slightly different. Therefore, the assets will become part of the decedent’s estate.
The estate is required to pay its debts before distributing assets to heirs. This is the same as if the decedent had a home subject to a mortgage. The heir might inherit the home, but the home would remain subject to the mortgage debt and lien. The heir does not become personally liable to pay the mortgage, but the mortgage would almost certainly have to be paid in order for the heir to retain the home.
The answer, then, is that you do not become liable for the debts of a business which you inherit.