The best evidence that sleep is important is the inability of humans or animals to voluntarily stay awake for long periods of time. Eventually sleep will overcome even the most motivated attempt to stay awake. This is an indication of an intense biological drive for sleep which is present in all species studied to date. Exactly why humans and animals have this drive is not well understood, but it clearly indicates that sleep is important, not dispensable.
In experiments conducted with rats, deprivation of all sleep or just REM sleep resulted in death of all animals within a few weeks. Thus, we know that sleep serves a vital function in animals. Since these types of experiments are not conducted with humans due to ethical concerns, we cannot say with certainty that humans would die without sleep. However, we do know that reduced sleep in humans consistently is associated with impaired immune function, metabolism, and control of body temperature, plus reduced learning and memory and mood control. Thus, it seems evident that sleep is important for proper function of many body systems and behaviors.