Is a business obligated to honor gift certificates issued by the previous owner?

It depends on how they bought the business. To oversimplify somewhat, there are essentially three ways to buy a business:
1) The business was an LLC or corporation and the new owner bought the LLC or corporation itself. Since the business is the same legal entity, it has to honor gift certificates, etc. pursuant to their terms; the fact that a new person (or other LLC or corporation) now owns the LLC or corporation does not change any of its obligations.
2) The business was an LLC or corporation, but the buyer did not buy the LLC or corporation--the buyer bought basically everything (including, say, the name) the business owned and hired some or all of the staff, but it's a new and different legal entity. In that case, the business does not have to honor the gift certificate, the same way that if you had an agreement with a friend that he could use your car as long as he bought case for it, but you then sold the car to a new owner, the new owner does not have to honor that arrangement.
3) The business was not an LLC or corporation (for example, it was a sole proprietorship). In that case, there was no legal entity to sell: by definition, some new person or entity (e.g. an LLC or corporation) bought what the old business owned, same as 2), above. In that case again, since the new business is a different legal person than the old one, it does not need to honor the certificates.
So you need to know how the business was sold to answer this question.
Of course, even in case 1), even though the business should honor the certificates, if they refuse to do so, you'd have to sue them to force them to honor it (or get an amount of money from them equal to the remaining balance). It is highly questionable whether it's worth suing for $69.03.