The poem uses alliteration to link concepts and words together. The speaker is stuck in a perpetual state of "unrest" caused by his love sickness. It's a state that is cyclical, with the speaker craving more of the very thing that makes him sick, thereby making him more sick, which leads to more craving! Repeated sounds like alliteration thus suggest the presence of a repetitive pattern of behavior.
This effect is especially clear in the opening two lines:
My love is as a fever, longing still
For that which longer nurseth the disease;
"Love" is linked to "longing," which in turn joins up with "longer." The speaker's love creates his longing which only prolongs his suffering, which makes him feel his desire more strongly... and so on until infinity, or death. The slippery, liquid /l/ sounds throughout these lines also subtly evoke sexual intimacy.
In line 8, the speaker says that "Desire is death." Here, too, alliteration joins the two concepts together. This is the speaker's point—that desire and death are inseparable. The /d/ is a dull, thudding sound, befitting the mention of being dead.
Line 9's alliteration pops up alongside parallelism, as the speaker says he's "past cure" and "past care." In other words, it's too late for the speaker; his love sickness is here to stay, and he's in so deep that he doesn't even care. The alliteration draws attention to these phases. There's only one vowel of difference between these phrases, making them sound almost like gibberish—the ravings of a madman (which the speaker feels he is becoming). Line 11's alliteration has a similar effect, linking the speaker "My" with his own perception of his madness ("madmen's").