Good question. As some have mentioned, I think the estate would be responsible for paying those outstanding bills, which then means (at least in my way of thinking) that if I inherited the estate, I would have to pay them.
But I'm no lawyer...I do know that the beneficiaries have to pay the estate taxes (death tax) that comes when a parent dies and leaves things to heirs.
When my father died he had a number of small bills which I took upon myself to pay, not wanting to have his creditors end up thinking ill of my father - or his survivors.