Does the Lord discipline with disease, and if so where in the Old and New Testaments are examples of him doing this in respect to us His servants?
To begin, it is very clear that the Lord does discipline His children (please see Hamartiology: the Biblical Study of Sin, section IV.5, "The Fact and Purpose of Divine Discipline"):
As many as I love, I reprove and discipline. So become zealous and repent.
As Hebrews 12:4-13 makes abundantly clear, if we never receive discipline, then we are not really the sons of God. Given that we are physical creatures with a spiritual part (although the day will come when our bodies will be exponentially more in tune with our spiritual side than is now the case; i.e., we will be pneumatikoi instead of psychikoi: 1Cor.15:44), it would be odd indeed if we were to be exempted from discipline that directly affected our bodies. After all, the first and arguably the most effective discipline that we receive from our natural fathers tends to be corporal. Job's sufferings came in two waves and in two types: 1) first he suffered material loss and the emotional pain that accompanies said loss (including grief for the loss of his children); 2) next, he was stricken with a debilitating physical illness. Now it is true that in reality Job was not being disciplined at all, but to both him and his friends, the idea of divine discipline assuming both of these forms (i.e., external material and internal physical) was, in principle, completely acceptable. Psalm 107:17-20 gives this principle of sinfulness ("folly", "rebellion", "iniquity") leading to physical complaints of the most severe type. And we find divine warnings of disease for the same in Deuteronomy (Deut.28:20-22; 28:35; 28:58-61) and Leviticus (26:16; 26:25).
Throughout the Psalms, we also find the Psalmists recounting divine "affliction", and I would say that this is primarily of a physical sort (cf. Ps.88; 119:67; 119:71; 119:75), and David's recounting of the physical effects of divine discipline are particularly trenchant (Ps.32:3-4 and Ps.38 passim; Ps.51:8). Paul's explanation of the illness and death that had afflicted a portion of the Corinthian community for their cavalier attitude towards the Lord's supper (and failure to confess their sins before partaking of it) is a clear attribution of physical divine discipline for sinful behavior (1Cor.11:27-32). The apostles, of course, also had the delegated authority to lay on this sort of discipline both directly (e.g., Annanias and Sapphira: Acts 5:1-11; and cf. Acts 13:10-12 in the case of an unbeliever), and indirectly (e.g., handing a person over to Satan: 1Cor.5:4-5; 1Tim.1:20). This is certainly also what Paul is implying in his warnings to the Corinthians (e.g., 2Cor.13:2-10). Two prominent Old Testament cases which come to mind are king Asa (where it is strongly implied that the disease in his feet was a divine punishment: 2Chron.16:1-13) and king Uzziah (where it is easy to conclude the instant leprosy is of divine origin: 2Chron.26:16-21).
While I believe that it is irrefutable that God does use disease as one of His tools to warn us, to discipline us, and to chasten us (all three of which are somewhat different), it needs to be underscored that God is also the One who heals us:
Now see that I, I am He, and there is no God besides Me. I put to death and I bring to life. I have stricken [down with disease] and I will heal, and no one delivers from my hand.
So while God may smite us for a purpose, He can and does also purposefully heal us (as many of the passages above also make clear: see again Ps.32, 38, 51, 119). For God is the God who heals:
[The Lord is the One] who forgives all your iniquities, who heals all your infirmities.
Putting together both of these things in the above verse is important, because it acknowledges on the one hand that God does discipline us with disease for sin, but also that He heals us after we have been afflicted as well, once we turn back to Him (cf. Is.30:26).
Like Job, moreover, there are times when disease does have a divine origin but discipline is not the purpose. Paul, of course, besought the Lord "three times" to take away his particular "thorn in the flesh", but the Lord allowed this great believer to continue to suffer - for his own spiritual good and blessing rather than any divine discipline (2Cor.12:7-10; cf. 2Tim.4:20). Few of us, of course, even come close to being in the same class with Paul, but that does not change the principle that sometimes disease is a test, an exercise intended to build up our faith and to show us the sufficiency of God's grace even in circumstances where the world says everything is impossible. We are all tested in many ways, and I would doubt that there have been many believers in the history of the world who have persistently pursued spiritual growth and have not had to cope with tests of this sort.
Sometimes disease is just a natural part of life. Sometimes it is a consequence of past events of which we may or may not have had any moral control - like the reformed alcoholic who had become a believer and said when asked about his health problems, "God gave me a new heart, but He didn't give me a new liver". To this I would add by way of personal observation that the ankle I had crushed nearly 30 years ago was undoubtedly warning discipline from the Lord. It took me some time (and some more discipline), but I think I finally got the message. That doesn't mean I don't still jog with a bit of a limp.
Have we sinned? Then we need to confess our sin, repent of our sin, turn away from it and sanctify ourselves in our new life for Christ. That way, whatever further discipline or chastening may still be forthcoming is from a loving Father to a son or daughter who understands why they are going to the woodshed, and who can now accept and even take comfort in the experience as a demonstration of love (and even grow from it). Are we experiencing lengthy and severe chastening for some egregious and high-handed sins of the past (as David and many other have)? Then we need to make maximum use of the experience, using it to reinforce our commitment to stay away from sin and evil and cling to the good (especially to our spiritual advance and the accomplishment of our personal ministries whatever these may be). Are we (relatively) sure that the suffering we are undergoing is not disciplinary, nor a warning, nor an extended chastening? Then we should fight the natural impulse to despair and instead strive to rejoice in the power of the Spirit that God has found us faithful enough to trust with a test of this sort, and try to get the most out of the experience. But whatever the circumstances, the solutions and the means of coping are always spiritual, focused on God, looking to Him for strength, healing, refuge, and timely help.
All of the above listed responses to suffering of any sort (physical disease included) are, of course, easier said than done, and victory today will be challenged tomorrow for just as long as the testing/suffering/discipline endures. But there are certain very important things that we know as believers and can never afford to let slip from our hearts. We know that God loves us. We know that He sent His beloved Son to die for all our "infirmities". We know that He forgives sin. We know that He heals disease. We know that His grace is sufficient for whatever test/trial/disease it may be our lot to encounter if so wills the will of God. And we know that whatever the origin of the trouble we find ourselves in, the spiritual application is exactly the same in every case: to trust Him, to wait for Him, to pray, to be patient, to rejoice, to persevere - for in each and every case we know with the knowledge of faith that sees beyond this temporary and hateful material world that "our God will come", that He most assuredly will deliver us, that He is hearing our prayers, and that this mountain that stands in our way will move one day. We know it better and to a greater degree even than we know that we are even here in this temporary world, because we have committed ourselves to Him and His Son our Lord in total faith.
So no matter how we got "here", whatever and wherever the "here" of our current trouble and tribulation may be, we have to keep walking through the trouble in ever deepening faith, ever sharpening hope, and ever increasing love. The day will come when all this is behind us. Let us make it our hearts greatest desire to prove ourselves well-pleasing to our Lord in every way now, that we may reap the greatest possible reward to His good pleasure on that day of days. For it is in just such times that we demonstrate what we and our faith are truly made of.
There is much more at the following links:
Bible Basics 3B: Hamartiology: the Biblical Study of Sin
Confession of Sin
Sin and Forgiveness
Confession of Sin, Fellowship, and the Filling of the Holy Spirit.
Recovering from Sin
The Cross, Sin, and the Devil in God's Plan.
How do you Prove Sin to Someone?
Have I committed the unforgivable sin?
The unpardonable sin and blasphemy against the Holy Spirit.
Personal Sin in the Old Testament and in the New.
Addicted to Sin.
Confession of Sin in 1st John 1:9.
Does exceptionally sinful behavior indicate that a Christian has lost salvation?
Does Hebrews 10:26-35 ("deliberate sinning" etc.) mean that a believer can lose his or her salvation?
Categories of Sin in Psalm 19.
In Him for whom nothing is impossible, our Lord and Savior Jesus Christ.