A refrain is a line or set of lines that repeats at regular intervals in a poem. With that in mind, "Alone" by Edgar Allan Poe does not have a refrain, but it does have other elements of sound to support the tone of isolation and despair.
Alliteration is a poetic technique which repeats a consonant sound at the beginnings of words within a single line of poetry. Here's an example:
From the sun that ’round me roll’d
In this line, Poe intentionally drops the a from around to create alliteration, paired with roll'd. (Interestingly, he also drops the vowel from rolled so that the words are visually similar, bringing further attention to the similarities in these words.) Poe crafts this line to draw attention to the passage of time. As the sun is "rolling 'round" him, the years are ticking by. And with each year, he is left alone. Apart. Different. This therefore evokes a feeling of disconnected gloom.
Consonance is the repetition of consonant sounds in a single line—not restricted to the beginnings of words. Here's an example:
As others were—I have not seen
As others saw—I could not bring
My passions from a common spring
(I've included three lines here for full meaning, but any one of these lines alone has that s repetition.) Through the consonance in these lines, you can hear the speaker hissing his feelings about his separateness. He has always seen the world differently. He has always drawn passions from places where others didn't. And this journey has left him at times bitter, as heard in the harsh sounds in these lines.
Assonance is the repetition of vowel sounds in a single line. Here's an example:
From the lightning in the sky
The speaker finds great mystery in this incredible force of nature. It also brings attention to the fact that he has described his own life in much the same way: "Then—in my childhood—in the dawn / Of a most stormy life..." Assonance in this line is a verbal cue that the speaker identifies with those forces of nature which are wild, powerful, and even feared. It brings a tone of mystery to the sorrow that precedes these lines.
Poe also combines these sounds with others like anaphora (repeating the first words in successive lines, such as in the three lines in a row which begin with From) to create an overall tone of mysterious solitude.