Unless the lender (if the buyer is using a bank for a mortgage) requires it - there isn't any "responsibility" to one party. A buried oil tank is simply something a buyer can request the seller to, most likely, empty remaining oil, fill in buried with certain foaming chemical (by a DEC certified company), and put in a new tank above ground. OR the seller can tell the buyer that they can buy the house with the oil tank buried and try and push off the responsibility to the buyer to take care of it.
There's no rule or law that says anything has to be done with a buried oil tank. Now in terms of removal - this is something that if you request it, a seller will most likely NOT oblige due to the fact that it can cause many more problems. Any pressure tests, both negative and positive pressure tests can also cause a leak to occur so you're most likely not going to get any seller to have their tank tested or removed.
Filled yes - removed no. As a Realtor, I would advise my sellers to not remove or have tested their tanks. If a buyer pushes on one of these ends, I would simply advise my client to pass and wait for another buyer.
You'll find that even alot of attorney's are not well versed in buried oil tanks. It's one of those issues where there's no mandate for a residential owner to remove a buried oil tank.
So to answer your question - don't let a buried oil tank kill a deal on a house you like. Simply request that the oil tank be:
2. Filled by DEC certified Company - that will provide you, the new owner with a certificate.
And a new oil tank be put above ground. Request that when the work is done, that you are present and may work with the company to choose a location for the new tank.
In order to make it a SMOOTH transition - I recommend that you split the cost of the work.
Ultimately, you should consult your attorney to cover all your bases.