If you don’t eat for a week, but do drink water, salt, have vitamins and minerals, then:

• Your body temperature will decrease somewhat as your ATP production is reduced, brown fat stops generating excess body heat, etc.
• Partially damaged cells are digested and replaced sooner to improve efficiency. This can reduce senescence and other signs of cellular aging.
• Growth hormone levels increase. This is an adaptation to spare protein for tissue repair rather than allowing it to be converted into fuel.
• Your hunger is likely to go away after a couple of days, but when exposed to favorite food smells, you may be compelled by them. Varies by person.
• Your feeling of alertness and consciousness may change. Some people get a little dizzy, and others feel things are more crisp. Some people struggle with long term higher order thinking, and some are energized by it.
• Your peak physical exertion will fall, because your anerobic capacity will be depleted substantially. Glycogen stores fall to lower levels, and your body spares glucose wherever possible. Glucose is reserved for red blood cells (cannot use other fuels), and some types of brain cells, and otherwise is shuttled back and forth as an energy carrier through pyruvate and lactate cycles. Some of this will be replenished from glycerol liberated during fatty acid metabolism; however, during an extended water fast, you do not produce enough from that.
• Your body will convert some muscle tissue into carbohydrate to maintain glucose reserves. For a week, typically this is not substantial, and can be compensated for by normal strength exercise during your refeeding week. This accelerates somewhat for longer fasts.

It’s important to note that medical fasts over 1–2 days typically are not zero-calories. 2–4 day fasts are typically 200–300 calories, and fasts over 5 days are typically 500 calories (aka ultra-low-calorie-diet). The diets typically are specially structured to ensure adequate protein, specialty nutrients, etc, but are low enough calories to ensure all of the same benefits to metabolic functions of a fast. Zero calorie fasts can suffer refeeding syndrome when extended beyond 2 weeks, but 500 calorie fasts have been maintained for more than one year in the most extreme example. You may find 500 calorie diets to be enough to alleviate the uncomfortable side effects found with zero calorie diets, and can be a great way to work up to a water fast if that’s a specific goal you want.