Does heating water up then cooling it down get rid of chlorine?

I am not sure, but probably not. You used to be able to let in sit for a couple of days, and the chlorine would evaporate. There is a product I use for all of my aquariums called, "Stress Coat". It will remove chlorine and Chloramines (which is equally as deadly and has probably been added to your water). There are many products you can buy to de-chlor water, but believe me, they are not effective. To save a couple of bucks, I used cheaper stuff, and ended up killing my fish. You can get this stuff from Walmart for about $6.00. Or you may find it in a smaller quantity at a locat fish store for $4.50. Good luck