Is it possible to remove salt from food without changing the flavor? I love soup, even a thick soup for dinner, yet many soup products have 400 to 1000 mg of sodium in them. Is there a simple chemical procedure any consumer could do to take the sodium out of the product without 'hurting' the flavor? 

If one takes some, say, Campbell's soup and heats it to boiling, then freezes it, then heats it again, then freezes it etc., would this extract some sodium?
Short answer: no, there is no easy (or even moderately hard) procedure, and certainly not boiling and freezing.

The thing is that water dissolves sodium ions better than just about anything else, so almost everything will come out of solution before the sodium does. You're much better off with a product that starts with less salt to begin with.