It depends on the illness, but a cold is a fantastic time to fast. It often helps to drink at least veggie broth to provide potassium and other needed electrolytes that flood the cells and allow them to release acids.
when you have a fever, your body is desperately trying to heat up the stomach/small intestine to kill any new bacteria and burn up toxins/mucous - rather like a self-cleaning oven - but that often fails and the entire body heats up as a result. So ginger teas and other heating things - fennel is very gentle and awesome - helps that action.
When the body wants to heal and cleanse - and a cold is a cleanse - it doesn't want to have to digest things. It wants to purge. So only eating very simple things like broth is a great idea, one that our precedents knew well and we don't practice as much anymore.