Can narcolepsy cause brain cells to die?

Other way around. Actually, narcolepsy is caused by the death of brain cells, not the other way around. Some cells in the hypothalamus produce a chemical called orexin or hypocretin; this chemical in needed for normal sleep/wake cycles. In narcolepsy, these cells are destroyed by your own body (autoimmune), resulting in the loss of the chemical. Presently, there is no way to directly replace the chemical.