Question:
I have a question regarding what is meant in Isaiah 53 when it is said that Jesus "took up our infirmities... and by his wounds we are healed" (v. 4-5; cf. Matt. 8:17).
In Brazil, many health-and-wealth types of sects claim that that means, and I quote one of them, that Christians have "a God-given RIGHT" to be healed, to be healthy, "to have a healthy liver, heart, or kidney."
They also claim that all illnesses are from the DEVIL or are demonic (this is easily refuted by many Psalms such as Psalms 38 or 102:10 --- the afflictions come from God!), so if you have one you need deliverance.
If you fall sick you are not in God's grace, and if you remain ill, of COURSE that is because your faith is not strong enough. So you have to keep praying (and it can't hurt to pay the pastor more money to pray over you -- after all, the pastors are rich because they are "blessed" by God and spiritual blessings are always financial...)
The "biblical" basis for the "fact" that diseases, ailments, etc. are demonic, comes from the Gospel accounts where Jesus heals them and casts out demons. But while this is maybe sometimes the case, it's not always so.
What is the BEST way to refute this nonsense that diseases are all demonic influences or demonic possession and not an after effect of sin, original sin and the frailty of the flesh?
Dear Tim:
I am very sorry that it has taken several months to answer your question. This forum has been closed since 2005. I am not sure how you and others found the submit page, but I will have to look into it. That is why I did not notice your question until now. I just discovered this today...
Concerning your question:
Well, it is not likely that reason will change their minds. All you can do is to present them with the correct interpretation and refutation from other Biblical references as you have already pointed out.
The consequences of Original sin includes illness and death. While the Cross saves us from spiritual death, it does not save us from the physical conseqences of orginal sin. If we were to be totally healthy and not illness claimed us we would never die.
Before the Fall, before Original Sin, there was no illness and no death. That pre-Fall state will be returned to us -- someday -- in heaven; but not now.
The bottomline is that all you can do is "reason" with them with Biblical passages, theological arguments, and logic. Since they are inherently illogical -- don't hold your breath 
God Bless,
Bro. Ignatius Mary