Red John isn't afraid of death.

Season 4 finale;

PJ: "Go to hell"
RJ: "As you well know Patrick, there is no hell.When I die I won't be punished."
This conversation shows that RJ knows he is going to die sooner or later,still he doesn't care, and still he continues to live his life on the edge.

RJ kills for the immediate pleasure of the act, and he does it in a way to offer fear to the people.He is a SHOWMAN.
Killing is his business.Cutting his victims open with his knife, feeling their fear.He watches them die and he feels nice.So I suppose that his adrenaline is at the maximum when he kills, he feels pleased and his feeling the heat.
And remember the episode where Jane has fugue, there is a firefighter that is stabbed to death and his wife stated: "My husband was addicted in adrenaline.Definitely when he was stabbed, it was the greatest excitement of his whole life."
I believe same goes to Red John.When he dies, he feels it, he feels the pleasure.This is my point of view.Remember Dummar, end of season 1, he got shot by Jane, and he died laughing and saying "Red John". Now this is a proper way to die for a psycho.

Red John has phobias like everybody else.Phobias that may seem ridiculus to others.Still I can't see a good reason why RJ is afraid of dying.

And by the way, how many fears Red John has?
A)Fear of birds.Even Sheldon from Big Bang Theory learned to tolerate birds eventually.
B)Fear of heights.Very common fear.
C)Fear of death.
D)Fear that someone is better than him.(He kills to punish those who expose him just to prove he is better)