It is unlikely that your fish will immediately die but his life will be shortened quite a bit. Heaters maintain the water at a proper temperature. Without a heater, the temperature will fluctuate which weakens the immune system and leads to disease. The fish will also be to cold, which also weakens the immune system and is generally detrimental to your fishes well being. Bettas are tropical fish that are biologically designed to live in waters of 76-86 degrees, when they get to cold they can not function naturally.
How big is your tank?