The best answer would depend on more information, such as the type of debt and the relationship to the deceased, however, there are essential biblical concepts that apply to our Christian fiscal responsibility. We are obligated to demonstrate principles of truth,  righteousness, wisdom, justice and love as debtors to God for whatever we have.

In Philippians 4:19, The Lord promised to supply all our needs.  We are told in Romans 13:7 & 8  to "Pay to all what is owed to them… Owe no one anything, except to love". In 1 Timothy 5:8, we are cautioned "If anyone does not provide for his relatives, especially for his immediate family, he has denied the faith and worse than an unbeliever." 

Psalm 37:21 declares, "The wicked borrow and do not repay." If the deceased who incurred the debt was a believer there was an obligation to repay creditors to the best of their ability. According to Ecclesiastes 5:5- "It is better that you should not vow than that you should vow and not pay". King Solomon wrote that a believer placed in a position to guarantee or co-sign another person's loan (Proverbs 11:15) would "surely suffer".  When a loan commitment has been made, the satisfaction of Justice is required.

Under Israel's Sabbath and Jubilee cycles (Deuteronomy 15), debts of all kinds were cancelled and each family's inheritance restored.  They were admonished to "Give freely and unselfishly, and the LORD will bless you in everything you do… Give generously from what the LORD has blessed you with". While we cannot expect the world to adhere to these same principles, we must consider our personal responsibilities and obligations.

Jesus' words in Matthew, (5 and 6) implore us to give in secret, storing up "treasures in heaven"; that we should "give to those who ask of us",  owing one another "nothing but love".  In Luke 19, He commended Zaccheus for his honesty in restoring funds.

According to modern law, the estate of the deceased person is responsible for paying debts through an executor or administrator. An individual might be legally responsible for the debt if they co-signed a loan or are the spouse of the deceased and state law requires payment of a particular type of debt. Similarly, an estate administrator would be held accountable for  complying with probate law. If an estate cannot cover a just debt, it typically remains unpaid, but the creditor has a legal right to try to collect.  If it is established that the debt is true, Christians are subject to individual conscience and the law of the land to address the financial affairs of a deceased family member. 

The obligation may be difficult to satisfy under certain circumstances.  For instance, a Christian executor might have strong feelings about repayment of debts incurred by gambling or vice,  but even these would be subject to just and righteous satisfaction under the law.  Certainly, one would wish to avoid needlessly exposing the faults of the dead.

Micah 6:8 (NLT) provides the clue to managing all of our affairs: "… the LORD has told you what is good, and this is what he requires of you: to do what is right, to love mercy, and to walk humbly with your God".   We know from John 3:27, that "A man can receive nothing unless it has been given him from heaven", and we have been given the Holy Spirit to discern what is pleasing and acceptable in God's sight.