Basic General Adaptation Syndrome (GAS Theory) from Hans Selye tells us that as our body is stressed in some way, it adapts to handle that stress. The basic principle works for getting stronger and for other stressors like getting sick. The process of adaptation requires energy in some form--and this includes immune system response.
Now, if you have more than 20% body fat, your body will probably be OK pulling the reserves from your fat. People with less than 15% body fat will likely need to supplement with food. That only speaks toward energy. NOTE: if you want your body to pull from your fat stores minimize the amount of carbohydrates you have. Carbs cause an insulin response, which suppresses your body's ability to burn fat until the blood sugar is back to normal.
That said, your body needs other things than just food. You may need to increase your micronutrients to provide the raw materials for your body to construct the antidote for your particular sickness. This is where extra Vitamin C, magnesium, zinc, etc. help fight off sickness. Typically this is done with vitamin supplements.
Lastly, most of your recovery processes happen when you are asleep or at the very least in a state of relative rest. Two of the most important things you can do in recovering from sickness is stay hydrated and sleep.