In my semi-ideal world, the only transactions taking place at the Federal Reserve banks would be between a few conversion agencies and the government.
For simplicity imagine a single company called Bitaxes. You go to their site, fill out your tax forms on their site then at the end it has a dollar amount you owe (this would be the only time dollars ever even come into play anymore). You submit your tax payment to this company in the form of bitcoins and they convert those bitcoins to dollars and send a check in your name to the IRS through the Fed bank. All your taxes get paid and everyone is happy (except you because you just got robbed).
Then when the government goes to spend money, the government gives everyone their pay checks through those Fed banks but everyone has signed up with Bitaxes company to convert their paychecks to bitcoins (because nobody uses dollars anymore except the government).
All of this could happen with no change in laws or the government changing a thing.