The United States is "liable" for the Treasury notes and bonds it issues. It is not "liable" for FRNs and is not required to pay them.
Art. 1, § 9, cl. 7 of the U.S. Constitution reads as follows: "No Money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law." This means that all debt instruments on which Uncle Sam is "liable" must be authorized by some law enacted by Congress. This is the constitutional provision requiring what is commonly called the debt ceiling, and if the debt ceiling is reached, no more debt instruments may be issued.
The States have similar constitutional provisions and a number of cases regarding this issue.
You can read current FRB financial statements.
PDF from the Atlanta FRB. You can see that FRNs are listed as liabilities of those banks. Can somebody please cite for me the law enacted by Congress that makes FRNs debt instruments to be paid by Uncle Sam? This is one of the constitutional/statutory flaws regarding the legal basis of the FedReserve System.