These are a little tricky because the Korean verbs have more diverse usage.
1. 상하다 (vi, vt) - common word with wider meaning including "to be hurt" - more commonly intransitive.
2. 손상하다, 손상시키다 (vt) - formal word most often used for physical things, including the body.
1. 다치다 (vt, vi) = to receive injury (in a body part), to be injured.
2. 부상을 입히다(vt) = to injure (someone), 부상을 입다 / 부상당하다 (vi) = be injured.
- I have injured my voice = (저는) 목소리(voice)를 상했어요 / 성대(vocal cords)를 다쳤어요.
- My voice is damaged = 목소리/성대가 상했어요.
- H can't sing because of the damage to his voice = 그 사람은 성대의 손상 때문에 노래를 못 한다. (손상 used as a noun)
상하다, like "damage", works around the "thing" that is damaged, not the person. But it can be both "to damage" and "to be damaged", as in 위가 상하다 (one's stomach is damaged/ruined), 위를 상하다 (ruin one's stomach).
Likewise, 다치다 takes the thing as the object instead of the person when used transitively (저는 팔을 다쳤어요 = I injured my arm). When used intransitively, the subject is always the person (저는 자전거를 타다가 다쳤어요 = I injured myself wile riding a bicycle). If you injure another person, we say 사람을 다치게 하다.