So, the book Alcoholics Anonymous kinda implies that selfishness kills us addicts and alcoholics. Rude.
But anyway, does it have a point?
Firstly, if you're an addict or an alcoholic, you're f***en' nuts, at some ineradicable level, even if you're sober / clean / abstinent / vegan / super pleasant to be around and have been for years. One day, you're gonna think a drink (substitute your poison) is a totally great idea, and once the genie is out of the bottle ... So, to survive, you need to be on a leash. An actual, spiritual leash. God needs to be in charge, so, when that moment comes, the Higher Authority yanks little lamppost-sniffing you away from the toxic goodies.
To enjoy God's protection in that way, I need to be serving God.
If I'm busy serving self, I'm not serving God.
So, if I don't get rid of self, I can't be protected by God. QED already. BUT:
Secondly (there's always a secondly): let's look at the results of selfishness itself. Running after sex, money, power, prestige, comfort, thrills, and appearance. This is tiring. It also induces fear, frustration, disappointment, and despair. All negative mental states have physical correlates in the body. Look around. People. Everywhere. Not exactly glowing are they? Modern society: not so well, in great part. There are exceptions. Go count them.
But thirdly (I'll let it go at thirdly): life is not existence. 'Killing us' is killing the life within us. Heaven forbid that an existence be devoid of life, but it happens. Existence is the vessel for life, which is vitality. Selfishness kills that and leaves us with existence. No one ever wants to terminate life. Life is health, happiness, harmony, love, joy, peace, and connection. No one ever has a problem with those. People often want to terminate existence. Why? Because once the bottle's empty, if you don't know it can be filled, you throw it away. Perfectly rational.
Conclusion: selfishness kills in all these three ways.