What happens if the dorsal root of a spinal nerve is damaged?

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.
Pain, burning. The dorsal root ganglion is a very sensitive part of the nerve origin and may be damaged. It results in severe pain, burning, tingling, numbness, and weird sensations such as a feeling of bugs crawling upon the extremities (called paresthesias).