How do I determine the temperature at which a dissolved substance evaporates?
First, it is a good idea to clarify the meaning of the word vaporize from here:


  
  (Physics / General Physics) to change or cause to change into vapour or into the gaseous state
  to evaporate or disappear or cause to evaporate or disappear, esp suddenly
  (Physics / General Physics) to destroy or be destroyed by being turned into a gas as a result of extreme heat (for example, generated
  by a nuclear explosion)
  


Chemists should be careful when using this word because it can be used to describe evaporation, sublimation, and boiling.  Note the third definition, which suggests that the vaporization of matter typically involves a large amount of energy.

Since the mixture you refer to in your question is in a liquid state, I am going to focus on liquid boiling.  (Note, we can say that boiling is the rapid vaporization of a liquid.)  The boiling point of a liquid is the point at which the vapor pressure of the liquid is equal to the surrounding pressure.  The boiling point of a liquid is affected by intermolecular forces such as hydrogen bonding, dispersion and dipole-dipole interactions.

When a solute is dissolved in a pure liquid (your example of caffeine dissolved in water), these intermolecular forces are altered.  The end result is an elevation of the boiling point.  This change can be predicted quite accurately using concepts typically covered in the first year of a college-level general chemistry class.

So, we know now that a mixture of caffeine and water will have a boiling point higher than pure water; however what happens as the solution continues to boil?  I ignored an important point in the previous paragraph; boiling point elevation due to addition of a solute will only occur if the solute is nonvolatile. Caffeine is such a solute.  Therefore, as you boi your mixture, the water will be turned into a gas, but the caffeine will be left behind.  The volume of solvent is decreasing; therefore the solution concentration is increasing.  Once the caffeine solubility limit is reached, it will start to crystallize.

When you are left with a mound of caffeine you can continue to heat it.  As you heat it above about $230\,^{\circ}\mathrm{C}$ it will start to sublime, or in more general terms, vaporize.

For those interested, the experimental procedure for the extraction and purification of caffeine from tea leaves is described at an undergraduate level here.