Can a person die or heart stop while on life support?

Yes it is possible: Being on life support, usually means that the person is on a ventilator(breathing machine... which takes over the function of the lungs). But you still need a beating heart to survive. If the Heart stops working, the person would die within a few minutes

Limits of support: Life support comes in many forms.You can assist fluid needs & nutrition, which are seldom called life support but remain part of the process.You can give antibiotics to fight infection & meds to keep the blood pressure up.You can assist breathing,the mainstay of what is considered life support.These measures can give the best chance for survival but cannot maintain life when the illness ends it.