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: 
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!
Think of Taylor Swift who spends long days awake making songs and meeting fans, yet she still sings to this day.
Yet even Lady Gaga had trouble sleeping at times, and she still sings.
It's literally dependent though good health's important even for singing, like said here.