I have been an elder in a church in the Dallas area for nearly 25 years, and on a number of occasions the elders have gathered to pray for someone who is seriously ill. Our primary passage of Scripture is James 5:13-16:
13 Is anyone among you suffering? He should pray. Is anyone in good spirits? He should sing praises. 14 Is anyone among you ill? He should summon the elders of the church, and they should pray for him and anoint him with oil in the name of the Lord. 15 And the prayer of faith will save the one who is sick and the Lord will raise him up—and if he has committed sins, he will be forgiven. 16 So confess your sins to one another and pray for one another so that you may be healed. The prayer of a righteous person has great effectiveness (James 5:13-16, NET Bible).
Over the years we have seen a few instances of marvelous healing, and in the other cases, the outpouring of God’s grace so that the illness was received as God’s will for their lives.
You may want to look at the message I did on James 5, which deals with this text.
It can be a very wonderful time to gather together and pray for one who is ill. The important thing is to emphasize God’s goodness and His power, as well as to give the one who is sick the opportunity to confess any hidden sins (if this happens to be the reason for their illness - see also 1 Corinthians 11:30). If God chooses to heal them, they should gladly accept this; If God chooses to take them home, they, like Paul (Philippians 1:21) can eagerly look forward to seeing Him face to face.
You may also wish to look at this message in Philippians.