You've probably heard of the saying "starve a fever, feed a cold". It's an old saying that goes back hundreds of years to ancient medicine. The idea is that if you're hot, you need to put less fuel into your body so you'll cool off. However, if you're chilled or weak, you need to eat to stoke up your internal fires feed a cold.
But the starving part isn't really a good idea. When you're sick, your appetite often decreases because food doesn't taste as good and you're tired. Fighting off a cold or flu demands energy. You're best off eating light, healthful food whole grains, veggies and lots of water.