Feed a Cold, Starve a Fever

Going hungry or overly full will not speed the rate at which you get rid of a cold or reduce a high temperature. 

Consciously "starving" a fever is a particularly bad idea. For every one degree rise in body temperature, there is an estimated 7% increase in basal metabolism - the rate at which the body burns calories to carry out basic functions like breathing and pumping blood through the arteries. It's not that the body will end u in a severe calorie deficit after a few days of fever, but consciously withholding food if you're hungry will only make it harder to fight illness.