Who is responsible for faulty dental work that was done prior to the business being purchased by new owners? Are the new owners responsible?

It depends, among other things, on what form "the business" takes. As a general matter, if a dentist purchases the assets and customer list of another dentist, he does not become liable for the malpractice of the previous dentist. The result mi be different if a different form of business is used. See an attorney.