What is the effect of religious faith or lack thereof in a person's recovery from an illness? Is there any scientific proof that it has an effect?
Jesus is recorded as saying "according to your faith, so be it unto you." At times people have been healed miraculously as a "help" to their unbelief, and Jesus said during his earthly ministry "believe on me for the works' sake (what miracles you have witnessed)..."
My own experience as once an atheist but at 22 becoming a born again believer is that if we take Jesus at his word (the Bible), we will find that his healing power is the same today. I was blessed to find an excellent church that believed in divine healing and practiced "anointing and laying on of hands" for healing and for deliverance when I was just a new believer. This church was a full-gospel holiness church and the Lord was very present in it. Everyone was healed all the time, and in general no one went to doctors. So as a result, I saw divine healing all the time. My testimony to this day is that Jesus has healed me for 40 years without any help from any other source -- and its not that I haven't experienced sickness or injury -- but Jesus has miraculously healed me every time.
As for scientific proof, yes there is proof -- and many times people have been miraculously healed and even their doctors have been astounded -- "at the tumor that was no more there" or whatever the case was.
A women in her 50s with a severe case of stomach cancer was sent home by the doctors to die -- there was nothing they could do any more for her. She came to our church (this is 35 years ago), and she was prayed for and instantly healed and could finally eat just as any one else. She lived for 20-25 years more. There are many cases of being healed by faith in Jesus. Smith Wigglesworth -- a preacher who had an amazing healing ministry -- has a number of books on "ever increasing faith" and a collection of his sermons and an account of his life is also written by another author. His doctrine matched his healing -- he is a great preacher on "holiness." And that's the kind of life God will honor with healing for it is impossible to have faith where there is an abiding life of sin.