Whether or not the estate has an obligation to pay the mortgage is really dependent on the terms of the estate plan and the solvency of the estate. The fact that someone is on the deed to a property (whether a deed of gift or a transfer on death instrument) that automatically passes upon death of the original owner to a relative has little or nothing, really, to do with the estate plan. While it may have been part of the person's ultimate plan for the disposal of their assets in life and at death, that is different than being part of the actual estate plan, which deals with the disposition of assets upon death, setting out the wishes of the deceased as it pertains to all property.
There are lots of people who end up in this position, even though the original person on the deed did not intend to purchase or leave the other any property. It happens a lot when say a child does not have adequate credit to secure a mortgage to buy a home, but has the money to pay a mortgage. So, in that scenario usually a parent or grandparent will put the mortgage in their name (the other will live there and pay the mortgage) and then in the event the "helper" dies, they have it pass to the other at death, either thru a "TOD" or a "joint tenancy", so that in the event they die intestate, or if the will is challenged, there will be no question who owns the house...the equity in it, anyway. It (the instrument) is its own separate entity, not subject to the will except to the extent the deceased makes it subject (I'm getting to that part). Like a life insurance policy, that pays on death but is not subject to any terms of the will, it can stand alone since deed is its own instrument, separate from any wills or trusts.
Under federal law, the mortgage must be allowed to remain in effect without changes when it passes from one person to another because of a death. This negates any due-on-sale clause in the mortgage.
Who pays for the remainder of what is owed, however, generally depends on the deceased's will.
The will might stipulate, for example, that the heir receive the home, free and clear, with the estate to pay all monies due. In this case, the executor will either use liquid assets, or sell other non-liquid assets in the estate to pay off the mortgage.
Other times, it will say that the heir get the equity, plus X amount of dollars, which the heir can put toward the home's mortgage or not.
If the will is silent on the issue, it is the responsibility of the heir/person to whom the property transfers (they needn't even be an heir) to pay the mortgage upon its normal term due, or sell the property to satisfy the debt. In the event the property is worth less than the remaining mortgage, the bank will usually take the house in "deed in lieu of foreclosure" rather than seek overage from the estate.