Alliteration appears in nearly every stanza of the poem. It adds to the musicality of the "Song" while adding emphasis to key moments of the speaker's impassioned monologue.
For example, the bird metaphor in the second stanza is full of alliteration:
Its wings are wet with tears, its beak with blood.
'Tis a strange fowl with feathers like a crow:
The repeated /w/, /b/, and /f/ sounds draw out the speaker's heightened emotion, almost as if she's sputtering with hurt and indignation. Elsewhere, alliteration adds vehemence to especially scornful phrases and passages, such as "Dancing like demons" (line 29), "folded by a fool" (line 45), and "fairer [...] Fresh [...] folded" (lines 50-51).)
The device serves other purposes, too, as in the leaf metaphor of lines 26-27, where it highlights the contrast between the leaves that are "running" and those that "remain." (In other words, between the love that the unfaithful partner no longer feels for the speaker and the love he does feel.) And in lines 34-35, heavy alliteration mimics the sound of a "bell":
Their tongue was like a mellow turret bell
To toll hearts burning into wide-lipp’d hell.
Thanks to those emphatic /t/, /b/, and /h/ sounds, readers can almost hear that tolling!