What will happen if the dorsal root of spinal cord is injured?

You would lose or have altered sensation in a part of your body. How much change of sensation depends on how much of the root is injured. Where depends on which side and what level of the spine’s dorsal root is injured.
The dorsal (aka “posterior” aka “back”) root of your spinal cord carries sensory information from your periphery (i.e. skin) and relays it to your central nervous system (brain and spinal cord).
Specifically, the 2 dorsal roots (left side & right side) present at each level of your spinal cord are made up of neuronal cells. Where their cell bodies bundle together is large enough to appear as a budge. This is the dorsal root ganglia – “ganglia” being defined as a collection of neuronal cell bodies.
Axonal fibers from this dorsal root ganglia go in both directions:
1) proximally into the central nervous system at the dorsal horn of grey matter in the spinal cord and
2) distally to the respective band of skin that the dorsal root is responsible for collecting sensory information about the environment from (aka its “dermatome “, more on this below).
Follow the red line to trace a pain signal from the index finger to the dorsal root to the dorsal horn, and onward to the brain after crossing synapses and sides in the spinal cord:
A fully cut dorsal root would eliminate feelings of touch, pressure, vibration, pain, temperature, positioning in that side of your body at the respective dermatome. A partial injury may cause changed sensation in any of those dimensions due to signaling gone awry: burning, tingling, partial numbness, aka neuropathic pain.
Voluntary motor control is retained because this is conducted through the motor neuron pathway in the ventral (aka “anterior” aka “front”) root, but coordination will be much worse because you won’t be able to feel where your limb is.
Here’s a dermatome chart showing what level of the spine is responsible for sensation in what parts of the body:
An interesting example:
If your left dorsal root at spinal level C8 is injured, among other areas of your back and arm being affected, your left pinky, the far half of your left ring finger, and the side of your left hand opposite your thumb would all feel numb.
This is what feels strange and tingly when you hit your “funny bone” in the back of your elbow. What’s happening is that you’re hitting your ulnar nerve which is one of the several nerves in your arm that feeds sensory information from your hand via the dorsal root at level C8 (the ulnar nerve also feeds into T1).