Which part of the hand goes to sleep, will tell you what nerve(s) are being compressed.
The median nerve usually gets compressed at the wrist and the ulnar nerve can get compressed at the wrist and/or elbow. When we sleep we usually bend up our elbows and put our hands under the chin or behind the head. These positions, when held for long periods at time, compresses the nerves and the area that they supply, goes to sleep.
Classically the median nerve supplies the palmar side of the thumb, index, middle, and radial (thumb-side) half of the ring finger. The ulnar nerve supplies the palmar side of the ulnar (small-finger side) of the ring finger, the small finger, and the ulnar dorsal (back) of the hand. The rest of the dorsum of the hand, the part on the thumb side, is supplied by the radial nerve.
With the burning sensation at the elbow, you probably have compression of the ulnar nerve, behind the elbow. The ulnar nerve travels in a covered groove on the back, medial (inside) of the elbow. When it is struck sharply, an electrical shock can sometimes be felt to the small finger. Sometimes called the "funny bone".
But, you could also be compressing the median nerve at the wrist, since you feel the whole hand is asleep.
The compression of the ulnar nerve at the elbow is called cubital tunnel syndrome, at the wrist is called Guyon's canal compression. Compression of the median nerve at the wrist is called carpal tunnel syndrome.
There is also a possibility that it could be coming from the spine. But, it would require that you have compression of three cervical nerve roots (C6, C7, C, for the whole hand to be involved.
If the symptoms go away in a few moments after waking up, then surgery is usually not needed. Some patients, who a awakened all the time by the numbness and pain, can wear splints at night to keep them from flexing the elbows and wrists. This often takes care of the problems. A small, light weight wrist splint for the median nerve is all that is needed, it does not have to be a huge splint used for broken wrists. As to the elbow, a big, bulky, hard elbow splint usually keeps patients awake. All that is needed is something to keep you from sharply bending the elbows all the way up. This can actually be done by taking old tube socks and cutting the toes out to make a tube. Usually, three or four thicknesses is all that is needed. You can still bend the elbow some for comfort, but the bulkiness will keep you from really bending the elbows all the way up.
If the pain and numbness become persistent, you just can't get the sleep you need, or you are concerned about the problem, you should then see a hand or upper extremity surgeon.