Just like the title says, how dangerous would it be for the inhabitants of a planet if the moon had rings? Assuming that the planet is roughly the size of Earth and the moon is a little bigger than ours. Would the inhabitants risk annihilation by asteroids from the rings?
Not At All, For Two Reasons
First, if a moon's ring system extended far enough to overlap the location of its parent planet, the rings wouldn't form in the first place, since no ring can form where a planetary body sweeps them out (part of why Saturn's moons don't fall within its ring system). So the rings wouldn't rain down on the planet, because if they were in a position to do so, they would never have formed.
Second, rings are sparse things. Saturn has glorious rings, but their entire mass is 40% that of Mimas - or about a millionth the mass of Earth. It's a lot of mass, in human terms, but in planetary terms, it's very little mass spread over a very large space. The chunks of matter in those rings average about 10cm.
A moon with about the mass of our moon (maybe a little more) would have a much more tenuous ring, comprised of dust. So even if the host planet didn't sweep its orbit clean before any occupants could be endangered (it would), a moon's ring system would not present any peril.