I've heard that one of the purposes of sleep is to process adenosine that builds up from metabolic processes, and it makes sense that the effect of adenosine as a neurotransmitter is to produce drowsiness so that you are put in an unconscious state where adenosine can get processed. But why does adenosine require you to be unconscious to be processed in the first place?
ATP (adenosine triphosphate) is used by the cells as a source of energy. When you are awake, the brain cells use a lot of energy, and ATP is quickly consumed, and it becomes AMP (adenosine monophosphate).
One decay product of AMP is adenosine. So, the more your "fuel" is running out, the more adenosine has accumulated. So high adenosine is a signal for the body to sleep.
When you sleep, the activity in the body, and in particular the brain, is much lower, and is redirected from active functions (thinking, moving, eating, etc.) to passive function, and in particular regenerative functions. This is an opportunity for the body to concentrate on stockpiling ATP, and clean up adenosine.
To add additional information to the other answer, adenosine (as a drug) does not require anything for metabolism other than enzymes typically present in blood. There are some genetic factors that interfere with this metabolism, and from personal practice it makes for a rather uncomfortable moment.
Also, adenosine is not a neurotransmitter (directly signaling another neuron to cause depolarization), but it does inhibit neurotransmitter release (and is therefore typically referred to as a neuromodulator in this capacity).
Furthermore, adenosine triphosphate (ATP), as noted in the other answer, is quickly consumed. The body, without normal processes ongoing to replete the stores of ATP, will very rapidly run out of ATP (in the most vital tissues to life, such as brain and heart, within a matter of minutes). Without ATP, life is not possible. This is actually the process involved with what we call rigor mortis. When muscle cells run out of ATP (which, interestingly, is required to relax muscle tension) the body of the deceased person stiffens and remains so until the muscle proteins involved in contraction are degraded by enzymes.