Drink to me only with thine eyes And I will pledge with mine (1-2)
The opening lines of the poem have a lot of interesting rhymes. The first vowels of both "eyes" and "mine" rhyme, which suggests that the speaker's and Celia's eyes are somehow connected or identical. Something similar is expressed with the rhyme on "thine" and "mine."
Or leave a kiss within the cupAnd I'll not ask for wine (3-4)
The fact that the speaker only needs a kiss suggests he is trading a bodily need (for a drink or beverage) for something more emotional (a kiss, or a sign of love from Celia). This dynamic between mortal and divine, earthly and non-earthly, material and spiritual, dominates this poem in particular, and the collection from which it comes, The Forest, more generally.
The thirst that from the soul doth riseDoth ask a drink divine (5-6)
The speaker implies that his love for Celia is like a kind of spiritual "thirst." This suggests that love is like drinking a beverage. Is it just us, or does that not sound too romantic? In fact, it clashes with the poem's attempts to elevate the spiritual (love, the soul) over the material (wine, nectar, etc.).