In carrying out open market operations, the Federal Reserve usually buys and sells U.S. Treasury securities. Suppose the U.S. government paid off all its debt. Could the Federal Reserve continue to carry out open market operations?

Yes, the federal reserve continues to carry out open market operations. They further use open market operations to maintain the financial stability and keep a balanced budget in the economy. These alternative assets would have to be traded in deep highly active markets to avoid the federal reserve's actions causing distortions. The federal reserve can utilise its foreign reserves and other investments. Moreover, the government can change the tax rates to have sufficient cash in case it wants to carry out open market operations.