What will happen if an injury cuts the dorsal root of a spinal nerve that carries feelings of pain?

Numbness: The dorsal (posterior, toward the back) root carries sensation nerve fibers: feeling of light touch, vibration, sense of position (movement or where the arm or leg or joint is), pain. However, there is much overlap between the area serves by a single nerve, so there may not be total loss of sensation, depending on how the person is "wired.".

It may create: Neuropathic pain as this is in the vicinity of the dorsal root ganglion and may affect that nerve's functional distribution as well depending on what type of overlap or cross innervation that may be in that individual.