I think I have good news for you. I have never heard of a bank in the case of a residential mortgage having a "due on death" clause written into the mortgage to trigger an acceleration of payments and foreclosure. To me, it just wouldn't even make sense for a bank to want to include such a provision, increasing the risk of default. In other words, as long as the Estate (or upon distribution, the beneficiaries) continues to make the mortgage payments, the mortgage will continue -- and the bank should have no say in the matter.
In the meantime, your father should live and be well.
DISCLAIMER: THIS IS NOT LEGAL ADVICE NOR DOES IT CREATE AN ATTORNEY-CLIENT RELATIONSHIP
I AM LICENSED TO PRACTICE LAW IN GEORGIA ONLY. THIS IS ONLY GENERAL INFORMATION AND IS FOR EDUCATIONAL PURPOSES ONLY.
I too am unaware of a "due on death" clause that would require the mortgage to be paid upon the death of the owner and sole mortgagor. However loan documents almost always contain a "due on transfer" clause, so that when you transfer the home from your father's estate to yourself, the bank could accelerate the note and force you to pay the amount due and/or re-finance.
Even though the due-on-transfer clause is in existence, the bank does not have to enforce it and may choose not to accelerate the loan. However, I do not recommend making a transfer in violation of the due-on-transfer or due-on-sale clause. While for practical purposes, a bank may not exercise its due-on-sale clause/due-on-transfer clause, unless you have the liquid funds to pay the loan in full, you should not assume you will be able to refinance the property with the same or another lender.
In the case of a surviving spouse, I have seen banks allow the successor owner (surviving spouse) be able to assume the mortgage and "step into the shoes" of the original mortgagor. You could try to assume the mortgage, but since you are the mortgagor's daughter, not his spouse, you may have trouble having the lender agree to the assumption.
One thing to consider, if your dad is insurable, is purchasing life insurance to cover the amount of the loan in the event of his death. He could purchase the insurance, or you could purchase the insurance. Depending upon his age and health, this could be a way to eliminate your concern.
With regard to the loan, the only way to know the legal impact of your fathers death and any transfer is to review the loan agreement/deed to secure debt. I would recommend engaging an attorney who is licensed in NY and who specializes in real estate and/or lending to assist you further and give you specific advice with regard to this matter.
You should review the loan documents to see exactly what they say about whether the death of the borrower could invoke acceleration. It is also possible that state law might prevent them from forcing you to refinance, even if the documents do provide that.
However, even if they do contain language to that effect, it is unlikely that the lender or servicer would care so long as the payments are being made. However, there are some potential complications if you do that - since you are not the borrower of record, they will not ever be willing to discuss the loan with you if ever there is a problem, nor will they give you a payoff statement if you did want to sell or refinance, and the tax benefit of the interest payments cannot be claimed by you.
You might consider consulting a local estate and probate attorney. It might be helpful if you were to be appointed personal representative of your father's estate, since in that capacity the lender would have to deal with you in the same way as they would your father.
I don't understand why you would not want lower-rate loan interest? I am guessing you may be worried about having to get an even bigger loan, to pay off his debts if you have to go through probate to get the houe into your own name? Since you do NOT own it until Title Insurance says you do?
Sorry I am not licensed in New York, to give you a Bricks-&-Mortar answer.