The following thoughts are from a chapter about a Puritan named Anthony Burgess in Taking Hold of God: Reformed and Puritan Perspectives on Prayer by Joel Beeke and Brian Najapfour. There are many helpful points in the chapter, which focuses largely on Burgess’s teaching about Christ’s prayer in John 17. I found the following summary of Burgess’s teaching on praying for one another to be helpful:
“1) God has made you part of the body of Christ. If a part of your own body is injured, how does it affect you? You should have the same empathy for the body of Christ as for your own body.
2) God instituted prayer as a means to help others. Instead, we are quick to criticize each other. Rather than finding fault, we should pray for fellow believers. That is our duty.
3) Praying for one another will ease differences, jealousies, and suspicions. It will make the godly of one heart and one mind. If you find yourself thinking how poorly a brother has treated you, pray for that man. It will immediately ‘quiet those winds and waves.'” (Beeke and Najapfour, 103-104)