Can sunburn cause neck tightness and numbing sensation when you turn it?

It can. If the burn resulted in skin swelling (edema), you may experience a sensation of numbing or burning as the inflammatory process proceeds. This is quite normal.
Not really. It should not cause numbness or tightness. It occurs due to overexposure to ultraviolet radiation. Initially there is increased redness, then later pain, itching, scaling, blisters can develop. At times its equivalent to first or second degree burns; or could even be life-threatening. Please remember to use your sunscreen when ever you are exposing yourself to uv rays.