I don't know how many times I've heard someone trying to figure out if it was starve a fever feed a cold or feed a fever starve a cold.  Since I feel ill right now and I'm running a fever I decided to find out.  Well, if you google it, you'll find a plethora of explanations of where this saying came from.  For the most part I've found that it's starve a fever, feed a cold.  Basically if you have a fever, you don't want to add to your body's processing because you want to cool it down.

Either way you look at it though, all the medical sites are in agreement, it's folklore.  It's not a good idea to do either if you are feeling under the weather.  There's no reason to put your body through more stress by starving yourself or stuffing yourself.  Good to know.