The hot burner light on my Kenmore glass top range stays on all the time, is it dangerous to leave this alone and not fix it?
In the sense that it no longer serves its function (it's on all the time, so it's no longer an indication that you should be cautious of a hot surface,) there's some danger that you might get burned, or set down something that melts or catches fire when there actually IS a hot surface, since that's not otherwise visibly obvious.
Other than that, I don't think there's any danger to having the light on, as such. Just treat the thing as if the light was accurate, so as not to get burned when it is accurate.