Question: I have noticed that when I find myself sleep deprived my singing voice gets greatly impaired. It is difficult to reach and sustain higher notes and it's like my voice muscles are "sore". 

I guess that's a given, considering I'm tired, but I wanted to know what are the implications of these less than optimal nights in my voice. Should I avoid singing in this kind of situation? Can this permanently damage my voice? Can this be corrected by posterior well rested nights?

Answer: 
One word: Severely.

Of all the musicians in a group, the singer must be the most concerned about getting enough sleep. And the older you get, the more important this becomes.

You should also be concerned about staying hydrated, meaning: drink a lot of water. Be careful with alcohol or caffeine or smoking, all of which can fatigue the voice due to depriving the mouth and larynx of moisture, leading to fatigue and strain when you sing.

If you have had too little sleep and too little hydration, and you sing anyway, you will increase the likelihood of straining and injuring your voice by singing. So if there is a day that you have not had enough sleep, try to sing less and talk less also.

If you develop severe voice problems, the first thing a doctor will tell you is to stop singing and talking for a week, a month, or longer, and to stop rehearsing and performing until you have healed. You do not want to let things get that bad.

Some people are more prone to vocal problems than others. Simply be mindful of how you feel each day, and figure out what you should do to protect your voice from unnecessary strain. But of course you must also practice singing to keep your voice strong. You are like an athlete; you want to keep your physical training up, but you do not want to risk injury from too much exertion.

Your voice must last your entire life, so if you are serious about singing your best, be mindful of all the advice on this page.
The simple answer is that without sleep, your voice gets tired. Tired voices become injured over time. Tired voices are rested by sleep.

So, yes, you should avoid singing with a tired voice. You can extend your vocal stamina by singing a lot (which it sounds like you're already doing), and by avoiding tensing up when you sing (describing how is a little beyond the scope of this answer).

Most vocal strain is reversible; you can become hoarse and a few days later, you'll be fine. However, over a long period of time of damaging your voice, you can develop nodules on your vocal cords. These are permanent, but surgically reversible. That's the extent of my knowledge of vocal damage; I have no idea how long it takes to develop (but I suspect the time frame is years and decades), or any of the symptoms of having it, or how risky the corrective surgery is.

Here's hoping someone else can give you a more descriptive answer!