You need to consult a probate attorney in the state where he died, resided or owned land. If probate is in process, consult an independent attorney to review the probate documents and any business documents as there may be some form of buyout or other arrangement as part of the corporation.
The answer is, it depends. The nature of the business makes a difference, i.e. is it a professional practice or type of business that requires a special license? As for the tax debt, that also depends on how the will was drafted, when the debt accrued, and what the business company agreement states on the payment of taxes. These are complex issues that require examining many different documents.
Not enough facts. If the business is incorporated, it pays its own taxes. Can't tell if you are talking about income taxes, property taxes, or franchise taxes. No one can inherit tax debt of the decedent. However, estate assets can be required to be sold to produce money to pay taxes.
DISCLAIMER: This is not specific legal advice and does not establish an attorney/client relationship.
This gets complicated very quickly. It seems to me that the business has never been dealt with in probate. This needs to be accomplished and the status of the business needs to get settled.
The form of the business (proprietorship, L.L.C. corporation, etc.) can make a difference. In any case, unless the business utilizes pass through taxation, for example filing an IRS K-1, the taxes would likely be paid by the business.
If the business is not a separate entity and is a proprietorship then the business is basically just the person which means that when the person died the business died with them.
The bottom line is these issues need to be handled.