If in battle they run out of health, a 30 second timer appears and you have to rush over and spam A to revive them. This is a nuisance in boss battles and you can die while you're saving them.
I've never had a spirit die, so I don't know exactly what happens, but I think they drop some but not all of the dream pieces you need to remake them. I thought the spirits you get from AR were just rare, but I guess if they're exclusive to the AR cards then you're firetrucked if they die. Somehow I doubt the AR cards are reusable.