Most of Sonnet 60 suggests a deep connection between human life and the natural world, what with all its comparisons between human life in the natural world. Life is like waves, no wait, it's like a sun rising and getting eclipsed, no wait it's like vegetation and crops, and… you get the picture. But by the end of the poem, the speaker puts the emphasis on humans making things (like poetry) as a way of separating humans from the natural world, which makes them so subject to the ravages of time.
Shakespeare's poem arguably makes nature look like human life more than it makes human life look like nature. That's the speaker's attempt to make the world more meaningful than it actually is.
By portraying the downfall of the sun as "unnatural," or at least happening according to an extraordinary natural process, the speaker heightens our sense of outrage at the "injustice" of nature.