In Quantum Leap the character Sam Beckett frequently leaps into the place of people who are in dangerous, life threatening situations. What would happen if he were to be killed? Would he leap to a new person, leaving the displaced person to be the one to die, or would Sam die in the past and the displacee would be stuck in the future?
I'm not aware of any episode where Sam is injured in one body before leaping out. Such an instance would be the easiest way to answer your question, whether injuries stick or not.
According to Wikipedia, it's not entirely clear how Sam's body relates to the host body. He appears to have the abilities of his own body (walking even when the host has no legs), but the sensory input of the host body (feeling labor pains). The simplest explanation is that Sam's body is, in some sense, actually there when he leaps. If Sam were shot in his (invisible) leg while in the body of a legless man, the bullet would actually penetrate Sam's leg. It can't penetrate the host's leg, it's not there!
So my answer would be that if Sam was killed in someone else's body, Sam would die. Whether the host leaps back (and comes back from the dead!) or not (and is stuck in the future looking like Sam!) would probably depend entirely on the will of "God, time, fate, or whatever."
Always assuming whatever power leaps Sam doesn't intervene and leap him out first, leaving the host to die.
At the end of "Revenge of the Evil Leaper", Sam shot Zoe, the evil leaper, just as she was beginning to leap out of the body of the prison governor. The leap continued to its conclusion, and the real governor stood up, alive. Presumably Zoe ended up back in her own body in the future, dead. By that logic, any damage to a leaper when in someone else's body affects the leaper, not the host.