Alliteration has a way of weaving a poem together, working with its cousins assonance, consonance, and sibilance to create pleasant patterns. These devices are often a big part of what makes a poem sound poetic, different from everyday speech: part of the delight of poetry is that it draws the reader's attention to the sound of language, not just the meaning.
But alliteration can also turn the reader toward a poem's themes, and that certainly happens in "If thou must love me," where the alliteration mostly comes from the repetition of whole words, not just sounds. There are plenty of initial /l/ sounds here, but most of them come from the same word: "love." (See the Devices entry on repetition for more on this.)
The gentle, quiet /l/ sound that starts "love" often meets up with itself here, reflecting the poem's idea of "love's eternity": love reflecting love, forever and ever. But "love" also meets with "look" and "let" and "lose," bringing the speaker's interest in love into contact with her list of insufficient reasons for loving.
One of most pronounced alliteration in the poem appears in lines 11-12:
A creature might forget to weep, who bore
Thy comfort long, and lose thy love thereby!
Here, gentle /th/ and /l/ sounds weave in and out of each other, contrasting with the crisp /c/ sounds of "creature" and "comfort." These lines, coming right before the poem's volta (that is, the moment in line 13 when the poem turns away from its description of frail conditional love and toward eternal love), feels cumulative: this especially dense sound-pattern wraps up the speaker's argument, so that in the last couple of lines, she can focus exclusively on matching love with love again.