When taken out of a natural context, I have a tendency to believe that most love poems "sound" a bit on the obsessive side. Due to this, it is a difficult to read if this is the exact intent of the poem. Perhaps, love, itself, is a topic which lends itself to obsessive tendencies. Having said all of this, I think there are some lines in "Annabel Lee" that can reflect some aspects of an extremely consuming and all encompassing love. The closing lines to the first stanza might indicate as much, "And this maiden she lived with no other thought/ Than to love and be loved by me." This is an example of being able to read a specific couplet from a love poem in either way. The idea that the love interest of the speaker possesses "no other thought" other than the love of the other does reflect some aspects of obsession. The love spoken by the speaker is one that transcends her death and her absence, and the language employed reflects this:
And neither the angels in heaven above,
Nor the demons down under the sea,
Can ever dissever my soul from the soul
Of the beautiful Annabel Lee.
Indeed, these lines do reflect a sense of subsuming personal identity fully within another and could be perceived a sense of obsession. In the final analysis, the idea of the speaker conveying a love that is transcendent of human confines is both powerful and can be seen as obsessive and a bit on the "crazy" side. However, this only underscores the irrationality of love and how the subjective experience can be understood, but never fully felt and experienced outside of the personal.
In the poem's final stanza, the speaker says that he constantly thinks about Annabel Lee, and that each night he sleeps next to her in her toom. While this shows devotion, it also suggests irrationality. The speaker will literally enter the grave so that they may be together. Whether this fact makes his descriptions unreliable cannot be definitively stated, though it does create some question about his mental state.