What does the poet ask of Celia in "Song To Celia"? Why?

In "Song to Celia," the speaker asks Celia to drink to him, as in a toast, using only her eyes. He tells her that he doesn't even need a glass of wine for this toast; if she only leaves a kiss in the cup, he won't care whether there is a beverage or not. He goes on to explain that his thirst doesn't originate in his body or other natural cravings. Instead, his thirst rises from his soul, and he craves something "divine" in Celia.
In short, the speaker is smitten with Celia. It's hard to know whether this is love, and Celia certainly seems to have rejected him before, as evidenced by her return of his roses. But as evidenced in the opening lines, he still craves her attention. Even if she cannot toast him in words, he asks that she merely acknowledge him with her eyes. He wants to Celia to really see him, to give him some indication of her own interest.

In this poem by Ben Jonson, the speaker asks his beloved, the titular Celia, to "drink to me only with thine eyes." In this, he means that instead of drinking to him physically—that is, toasting him, or otherwise acknowledging him, with the use of alcohol—she should, instead, only throw him a look, and he will be sufficiently intoxicated by that. He, in return, will "pledge" his love to her with his own eyes and loving glances. Jonson is here referring to the longstanding tradition of using alcohol, particularly wine, as a sort of ceremonial tool of bonding. Using cups and wine has long been a way of pledging honestly, loyalty, and troths across many societies. In suggesting that Celia need only drink to him with her loving glances and the expression on her face, Jonson indicates that they do not need anything beyond their own love in order to know that they are sworn to each other.
He also suggests that she leave "a kiss" in the cup from which he might ordinarily drink, and if she does so, he will have no need for wine at all. Instead, her kisses will have the same effect on him, rendering him dizzy, relaxed, happy, and still more in love with her than he was before.