Can one preserve food by periodically heating it?
YES!  If food is quickly heated and kept above 140F/60C, microorganisms can't grow in the food.  Professional kitchens use steam tables to keep food warm at this temperature throughout a day's service.  If the product is cooled quickly (generally using an ice bath), then it can be used for more than one day. 

Another approach is often used for stocks and soups: heat them to 165F/75C daily or every two days, to kill microorganisms, then cool them quickly with an ice bath to under 40F/5C.  This can be done 2-3 times to extend the fridge life of the product.

The most important part of both approaches is to use a food thermometer to verify that food is fully heated/cooled, and to make sure that heating and cooling is done QUICKLY.  For food safety, you do not want food to be in the 40F/5C to 140F/60C range for more than an hour if possible, or two hours tops.  

Another trick to extend the fridge life of food is to immediately transfer it from a still-hot pot to a cool container, and then leave the top of the container open to allow steam to escape.  This greatly reduces cooling time; even in a refrigerator, a sealed container may take over an hour to cool down to a safe temperature.