According to Reformed Theology, why do unbelievers deserve punishment? I read this quote from John Calvin on some website:


  "...we allow that man has choice and that it is self-determined, so that if he does anything evil, it should be imputed to him and to his own voluntary choosing. We do away with coercion and force, because this contradicts the nature of the will and cannot coexist with it. We deny that choice is free, because through man's innate wickedness it is of necessity driven to what is evil and cannot seek anything but evil. And from this it is possible to deduce what a great difference there is between necessity and coercion. For we do not say that man is dragged unwillingly into sinning, but that because his will is corrupt he is held captive under the yoke of sin and therefore of necessity will in an evil way. For where there is bondage, there is necessity. But it makes a great difference whether the bondage is voluntary or coerced. We locate the necessity to sin precisely in corruption of the will, from which follows that it is self-determined. John Calvin from Bondage and Liberation of the Will, pg. 69-70


I believe in determinism, and I have no problem with the idea that humans need to be chastised for bad behavior. For example, a child steals a candy bar. The mother catches the child, makes them return the candy and apologize, and then issues whatever corrective chastisement to cause the child to think twice next time they want to steal.

I also have no problem with the fact that God appointed for this to happen, so that the child's will was not free. The child stealing and then getting chastised is a good lesson to both the child and others about the consequences of sin, and this lesson will cause them to behave appropriately (as well as continue the chain of determinism towards God's ultimate goal). Had the child never stolen, then the chastisement would have never occurred, and nobody would have learned any prudence.

What I don't understand is why the child deserves to be punished, namely in an eternal hell with flesh burning (or whatever other ideas are associated with this place.)

According to Reformed Theology, why do humans deserve to be punished forever in hell for their predetermined actions?
The simple answer is that God has decided that the wages of sin is death. It is entirely up to God as to what punishment sin deserves, and the punishment God chose was eternal damnation.

The Heidelberg Catechism's first section on Sin and Misery speaks about this, in particular question and answer 10:


  Q. Does God permit
  such disobedience and rebellion
  to go unpunished?
  
  A. Certainly not.
  God is terribly angry
  with the sin we are born with
  as well as the sins we personally commit.
  As a just judge,
  God will punish them both now and in eternity,
  having declared:
  “Cursed is everyone who does not observe and obey
  all the things written in the book of the law.”


The Westminster Larger Catechism also mentions this, in question 29 in particular but previous questions too:


  Q. 29. What are the punishments of sin in the world to come?
  
  A. The punishments of sin in the world to come, are everlasting separation from the comfortable presence of God, and most grievous torments in soul and body, without intermission, in hell-fire forever.


And also question 89:


  Q. 89. What shall be done to the wicked at the day of judgment?
  
  A. At the day of judgment, the wicked shall be set on Christ’s left hand, and, upon clear evidence, and full conviction of their own consciences, shall have the fearful but just sentence of condemnation pronounced against them; and thereupon shall be cast out from the favourable presence of God, and the glorious fellowship with Christ, his saints, and all his holy angels, into hell, to be punished with unspeakable torments, both of body and soul, with the devil and his angels forever.


And also question 152:


  Q. 152. What doth every sin deserve at the hands of God?
  
  A. Every sin, even the least, being against the sovereignty, goodness, and holiness of God, and against his righteous law, deserveth his wrath and curse, both in this life, and that which is to come; and cannot be expiated but by the blood of Christ.