Do I lose jump pay if the reason I can't jump is due to a work-related injury?

Generally yes, if you were injured training and it was not because of negligence then the commander sign off on you continuing to draw pay. He can also have it taken away from you, though I have rarely ever seen a commander do that for legitimate injuries. Either way, hold onto your profile for as long as you are assigned to that unit. The reason is because if your unit goes through a CIP inspection and they check the jump logs, they will ding the unit. If enough jump logs are deficient the unit may be required to start mandatory recoupment of jump pay for anyone who has an uncovered pay period and you have to prove why you were actually current (deployments, injury, TDY etc).

That being said, generally after six months you should be pulled off jump status and have your pay stopped if you still are on profile. I'm am not sure if this is a general commander's policy for airborne units or an actual regulation though.

What is supposed to happen is that a stop pay memo is submitted for you for each month you won't be jumping, that or you move onto Jump Rule 3 and you have 1 year to complete all 4 jumps for the year. There are more stipulations to it than just migrating to Jump Rule 3. What will most likely happen is that you will continue to draw jump pay until you ETS or PCS. When you clear finance they will go over your jump log and deduct pay for any months you were not current.

It doesn't actually matter why you don't jump, finance is only concerned with whether you jump or not. On the good side, if enough time has lapsed (more than a year) they often will not recoup the overpayment of jump pay. That's not set in stone, though.

You should be checking on this with your company operations NCO and your S-1 NCO (PAC NCO).