Report on the Federal Reserve's Balance Sheet August 2020 (40 pages):
Classification of Fed assets and liabilities taken from Table 1 on nominal page 4:
Assets = Securities + Repo + Loans + All Other Assets
Securities = Treasury securities + Agency debt securities + Agency mortgage-backed securities
Liabilities = Currency + Reserve Balances + Reverse repo + TGA + Treasury contributions to credit facilities + Other liabilities
To purchase any financial asset or extend a loan Fed simply credits the reserve balances of a bank for an increase and does not have to borrow or tax or save funds to do so. Repayment of debt owed to Fed, or sale of security or loan by Fed, Fed simply debits the reserve balances of a bank for a decrease.
Banks clear payment on behalf of non-banks so if Fed deals with a non-bank the banks still see an increase or decrease of reserves. However when a non-bank is involved in transactions with Fed there is an increase or decrease of bank deposits in addition to the corresponding change in bank reserves. So Fed can add or drain liquidity, in the levels of reserve balances (via deals with banks) and bank deposits (via deals with non-banks), by changing its asset position.
To have control over the monetary policy rate either Treasury or Fed can issue interest-bearing liabilities as alternatives to holding currency or zero interest reserve balances.
In terms of the original question the Fed purchased Agency MBS starting in late 2008 or early 2009 which perhaps were backed by the full faith and credit of the federal government because Treasury became Conservator of key federal home loan banks during the mortgage crisis. Fed could allow these assets to run off its balance sheet as repayment comes due and cancel reserve balances equal to the reduction of assets. Alternatively Fed could reinvest or further invest in Agency MBS. I cannot find a reference for Fed Agency MBS policy since 2008-2020.
During the 2020 Covid crisis Fed added a number of facilities shown in the assets of the paper on Table 1. These include a corporate credit facility, main street facilities, and municipal liquidity facilities, which may represent purchases of bonds in open markets to provide liquidity to the respective sector. Again when these repay the Fed would cancel the respective asset and corresponding cancel bank reserves. This policy would allow assets and reserves to run off the balance sheet. However Fed can reinvest which keeps assets and reserves at the same level or Fed can expand credit policy increasing assets and reserves.
The impact on the economy is complex and subject to economic debate under the concepts of monetary policy, monetary policy transmission mechanisms, lender of last resort policy, dealer of last resort policy, and federal government as investor of last resort since Treasury tends to put up equity funds to take a loss on programs managed by Fed.