While every American would say "not me" my mother always taught me the proper way to say it is "not I"
I did a bit of reading and it seems that both are correct now. Different people have different opinions about which one is "more correct", though. What's undebatable is that I is a subjective pronoun and me is an objective pronoun. It is easy to say which is correct in a sentence:
Clare and I are going for a coffee.
The dog followed John and me to the door.
But "not I/me" is not a sentence, so there's no subject and object. And now, some say that it shoul be not I, because it's short for I don't, while others claim that whenever subjective is not needed we use objective, very much like when you show sb yourself in a picture you say me.
Anyway, as I said, both are accepted and me is denifitely more common.