To answer this question you need to think about the structure of the poem that Marvell creates and consider how the poem is actually divided into three discrete sections, and how each of those sections are used to advance the speaker's argument.
The first section runs from lines 1-20, and consists of lots of examples of hyperbole as the speaker tries to describe the extreme lengths to which he would go to express his love for his beloved if he had but time to do so:
My vegetable love should grow
Vaster than empires and more slow...
However, in lines 21-32, the brevity of life is stressed as the speaker reminds his beloved that they do not have eternity to court. In reality, time is described as a "winged chariot hurrying near." The only future they have to look forward to are "deserts of vast eternity." Based on this, lines 33-46 move to the conclusion of the poem, as, with a tone of challenge and defiance, the speaker urges his mistress to "devour" time and "tear" the pleasures from life, because time cannot be made to stop. By so doing, the speaker argues, they can make the sun "run":
Let us roll all our strength and all
Our sweetness up into one ball,
And tear our pleasures with rough strife
Through the iron gates of life;
Thus, though we cannot make our sun
Stand still, yet we will make him run.
Thus each of the three sections of this poem is clearly built on upon the other and helps to advance the central argument of why it is important to love today and not wait for a tomorrow that may never come. "Coyness" is thus a "crime" based on the brevity of time and our inescapable death.