No, debt does not survive the death of the debtor. If the debtor had an estate, then creditors can collect money from the estate, but they can never collect from surviving relatives.
Collection agencies have been increasingly attempting to collect the debts of deceased people from surviving relatives. Often times, the collection agencies buy the debt of dead people because that debt is the cheapest kind to buy, often costing only one or two cents on the dollar. The collection agency will then call widows, children, brothers, and sisters of the deceased and say that the survivors have to pay the debts.
Survivors who were not co-signers or did not guarantee the debts of the deceased have no personal liability for the debts. That means that legally, the living have no responsibility to pay the debts of the dead. So what business do collection agencies have trying to make people debts for which they are not responsible? They do it to make money. People are sad and vulnerable after their family members die, and can be easily misled into giving money to creditors. Survivors often have a fresh payout from their life insurance policies, so they may be more willing to part with cash than normal. Debt collectors take advantage of people when they are at their most vulnerable to make money that they have no legal right to collect.
If you are receiving such phone calls, and sometimes they call ten times per day, make sure to tell the collection agent that you know you don't have to pay the bill and demand to be taken off their list. Furthermore, you should tell them that the Fair Debt Collection Practices Act (FDCPA) bans debt collectors from speaking with third parties without their consent. As a relative of the deceased you are a third party, and they should therefore not be talking with you.