Does desalination work?
This is a classic science experiment. Where saltwater (using $\ce{NaCl}$) is heated, as it does, the water evaporates and most, if not all, of the salt remains - increasing the concentration of salt.

Evaporating half the water would approximately double the concentration of $\ce{NaCl}$.

Eventually, the salt concentration becomes supersaturated and crystallises. The reason is due to:


  As the water evaporates less and less water molecules are present to keep the salt particles apart. The salt therefore recrystalises and can be collected.


Source

This process is also used in salt evaporation ponds, where


  seawater or brine is fed into large ponds and water is drawn out through natural evaporation which allows the salt to be subsequently harvested.


As Curt F mentioned in the comments, different salts and mixtures of salts can have different outcomes in terms of vaporisation.