Can you purify salt water with heat?

Yes, but in some sense you boil the water out of the salt!

By boiling the water, the water turns to steam and rising, but the salt does not, and so it stays behind. If you have a way to collect the steam and turn it back into water, you will have pure water! This is called distillation.

You can do the same thing just using evaporation. If you have water that is evaporating, but put a cool object nearby where the water can recondense, and then you collect the condensed water, the collected water will have no salt in it.

They even sell solar stills to do this. See the Web Link to left for more about this.