It's borrowed money. If it does not return to the Fed before you die, it's collected afterwards. Eventually, it goes back to the Fed, because it's theirs, not yours.
It's collected afterwards? When, and by whom?
If you're referring to taxes, doesn't that mean a gold-backed currency is also "borrowed money" since you still have to pay taxes with it?
It's an accounting system. There are no dollars in fact. There are debits and credits, and some debits are represented by those FRNs. If one is burned, or destroyed, the system adjusts itself, because at the end of the day, all accounts are "zero".
The fact that you have been left with a few of those pieces of paper is not an issue. One day they will be spent, and when they are spent, they are headed back to the Federal Reserve.
Second point, the gold question.
No, the congress, authorized to coin money, would use the gold to back the fiat currency, making it somewhat stable and insured. Currently, the public Treasury borrows from the private fed.
Wiki
A double-entry bookkeeping system is a set of rules for recording financial information in a financial accounting system in which every transaction or event changes at least two different nominal ledger accounts.
The name derives from the fact that financial information used to be recorded using pen and ink in paper books - hence "bookkeeping" (whereas now it's recorded mainly in computer systems) and that these books were called journals and ledgers (hence nominal ledger, etc.) - and that each transaction was recorded twice (hence "double-entry"), with the two transactions being called a "debit" and a "credit".
It was first codified in the 15th century. In modern accounting this is done using debits and credits within the accounting equation: Equity = Assets - Liabilities. The accounting equation serves as an error detection system: if at any point the sum of debits does not equal the corresponding sum of credits, an error has occurred. It follows that the sum of debits and credits must be equal. --
(Federal Reserve Keeps a Double Entry Bookeeping system)