The controlling principle of fire is that fire will keep burning until it's extinguished. As long as there is fuel a fire won't go out on its own. It has to be acted upon. Either you take away its fuel or you pour water on it. The same is true of sin. Sin won't "go out" on its own.
James uses the metaphor of conception and childbirth to describe sin's natural progression.
But each person is tempted when he is lured and enticed by his own desire. Then desire when it has conceived gives birth to sin, and sin when it is fully grown brings forth death. James 1:14-15
Sin will not only be striving, acting, rebelling, troubling, disquieting, but if let alone, if not continually mortified, it will bring forth great, cursed, scandalous, soul-destroying sins. . . . Sin aims always at the utmost; every time it rises up to tempt or entice, might it have its own course, it would go out to the utmost sin in that kind. Every unclean thought or glance would be adultery if it could; every covetous desire would be oppression, every thought of unbelief would be atheism, might it grow to its head.
Is Owen being hyperbolic? I don't think so. "Be killing sin or it will be killing you," he writes. These are scary warnings from the inspired apostle and the Puritan pastor. They should cause us to be more diligent in "killing sin". Sin's ultimate goal and outcome is death. The good news is that by the Spirit's power the child of God will win the fight against sin, and the Spirit's controlling principle is life.
For if you live according to the flesh you will die, but if by the Spirit you put to death the deeds of the body, you will live. Romans 8:13
Quote from Of the Mortification of Sin in Believers, pp. 52-3 of this edition