Heating or even boiling the water does remove chlorine from tap water. However, it does not remove chloramines, and many municipalities are switching from chlorine to chloramines. Chloramines have the same effect on beer as chlorine, forming chlorophenols which can taste something like someone spiked your beer with Comet cleanser. No bueno.
So, I don't bother boiling the tap water I use to brew. I add a bit of campden tablet. That removes chlorine and chloramines.
Sanitation is not an issue, since that water gets heated to better that 150F for quite a while and then boiled for an hour. Bye-bye, bugs.
Will ice get you? I don't know. I've never tried it.