Yes, the poem is a satire. Satire often uses exaggeration or hyperbole to poke fun at a weakness or foible in a person or society. In this poem, Marvell use hyperbole to poke fun at a woman who hesitates to give herself sexually to her beloved.
The speaker says if there were "world enough and time" the woman's coyness or withholding of herself would be perfectly fine. He could spend a hundred years admiring her eyes and two hundred years admiring each breast. He could spend 30,000 years singing her praises. This humorous exaggeration is trying to point out to her how silly she is being in his eyes.
As the speaker notes, humans are mortal and time goes by very fast. As he describes it, in one of the most famous couplets of the seventeenth century:
But at my back I always hear
Time’s wingèd chariot hurrying near
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Time’s wingèd chariot hurrying near
Yes, Marvell satirizes the social convention that young women must remain virtuous, i.e. virginal, until they marry. The speaker directly addresses one particular lady who he claims is behaving "coy[ly]": she is apparently rebuffing or refusing his sexual advances, citing her honor, and he hopes to convince her to sleep with him anyway. He says that "had [they] but world enough and time," her coyness would not be a crime; however, since all of us have limited time here on earth, and physical beauty is even more limited, she should cease her hesitation. He says, "at my back I always hear / Time's winged chariot hurrying near" and the future brings a loss of beauty as well as life. He merely wants to make the most of the time they have together. When they are dead, "worms shall try / [Her] long-preserved virginity"—why, he seems to ask, should she save it for them? Honor, he claims, will mean nothing then, and no one loves in a tomb. For now, she is beautiful and lovely, and if she were only willing as well, though they cannot stop time, the speaker implies, they can certainly pass it in more diverting and enjoyable ways.
Satire is the use of irony, sarcasm, ridicule, or the like, in exposing, denouncing, or deriding vice, folly, etc. (See links below for more information.)
The poem, To His Coy Mistress, is a satire in that it uses sarcasm to point out the fact that time is short, and to argue the point of the narrator that the young lady he is interested in should not deny his advances before time runs out on them. He starts out by exaggerating that if he had all the time in the world, he could take his sweet time and slowly woo her over thousands of years. However, the reality is that the clock is ticking and the narrator realizes that "time's winged chariot [is] hurrying near" and they need to speed up the process before death is knocking at their door. To compel her further, he comically states that, "The grave's a fine and private place, but none I think there do embrace."
In conclusion, the author uses sarcasm to denounce the young woman for being so coy with him, and uses the satire in his poem to politely build his argument for her to embrace him without reserve.
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