Not quite. A Federal Reserve Note is a debt instrument issued by the Federal Reserve Bank, which is as much "government" as the Post Office, Fannie Mae, or Amtrak--a tiny bit government, but mostly independent.
In theory, a Federal Reserve Note should be redeemable in United States Notes or in coins stamped by the U.S. Mint, but thanks to legal tender laws, the note is a debt instrument, which means it may be discharged by tendering Federal Reserve Notes in place of whatever the debt actually calls for.
So you go to the Fed and ask to redeem a $100 bill, and they will hand it right back to you. Technically speaking, they should give you a different $100 bill, other than the one you are trying to redeem, but really, if you get that far, you're already lucky they aren't physically throwing you out of the building.
This is one of those "laws and sausages" things. You're far better off not looking at it too closely if you want to continue to enjoy the benefits unburdened by unnecessary knowledge.
In practice I don't think it's useful to think about it in terms of liability. That $100 bill is a share in the net wealth of the nation. This is why when a country prints a lot if its currency to fund government spending, the relative value of that currency goes down. There are more 'shares' in the economy now, so each one represents a smaller slice of the pie than it did before. So if there is any liability, it's distributed across everyone else holding that currency.
Of course the value of anything is only relative to what anyone is prepared to pay for it, it's a matter of sentiment, so the value of an economy can go up or down depending on perceived risks, liabilities or opportunities and thus is reflected in the value of the currency compared to other assets or currencies.
If you don't believe that, try to pay your taxes with gold and see what happens.