Is cell membrane living
It is neither living nor dead. It is simply not alive the way that any other collection of chemicals is not alive. Is salt alive or dead?

A cell's membrane consists of a collection of chemicals (mainly lipids and proteins). Individual chemicals can never be considered living or dead, any more than rocks or water can be living or dead. While the actual definition of life is tricky, everyone agrees that in order to be alive you need to (at the very least)


Somehow interact with your environment
Be able to make more copies of yourself


Therefore, the cells themselves are alive but the individual parts of the cells are not since they cannot make copies of themselves (this gets complicated when considering some organelles but never mind that here). Similarly, you are alive but your teeth are not. While you can make more of you (you can have children) your teeth cannot make more teeth.