Medically, a week without food for a normal person of healthy body weight isn't smart but it won't kill you either. A healthy body can survive on water alone. It isn't pleasant, and the lasting side effects aren't a picnic either, but it's doable. That said, probably not advisable if you don't HAVE to do it.
The biggest problem, as mentioned already, is electrolyte balance. People who aren't eating tend to drink more; as you drink more, you dilute the electrolyte concentrations in your body. That will affect mental clarity and physical strength. You'll quickly begin to feel weak and will lose concentration easily. After 5-6 days, you can probably expect hallucinations.
You will also get hunder pangs. Gut-wrenching, vomit-inducing hunger pangs. Your body is used to eating on a regular cycle, so for the first few days they will be intermittent, but they may eventually become constant.
As for burning fat; the first few days of starvation will eat into muscle mass first, but not really significantly. Your body with then switch to fat-burning processes in combination with muscle degradation. As fat stores burn lower, your body will again switch to primarily muscle as a source of energy, and ketosis will set in (you breath will start to smell like acetone, or nail polish remover). That may or may not occur before 7 days, it really depends on the individual.
When you resume eating, your digestive system will be easily overwhelmed so you need to begin with small amounts of food regularly, and easily digestible foods - NO roughage (e.g. carrots, lettuce, etc). Breads, cheeses, and very small amounts of fruit.
Your electrolytes will be out of whack for a couple weeks afterward, as will your eating schedule and digestive system. You can probably expect diahrrea.
Not really a pleasant way to win a bet.
