With a love that the winged seraphs of heaven Coveted her and me. (lines 11-12)
This is the beginning of a sort of weird conspiracy theory that the speaker lets us in on. For some reason he thinks that the angels ("winged seraphs") in heaven were so impressed by his love that they wanted it for themselves, and so decided to snatch Annabel away from him. This doesn't sound like super angelic behavior to us, and it drives home the fact that Poe's world is often a pretty scary place to be. Most of us think of angels as comforting guardians who are above being jealous of people. For Poe, though, the world is full of dangerous forces, and heaven and hell and fate are all lining up to mess up human lives.
The angels, not half so happy in heaven, Went envying her and me-- (lines 21-22)
A little more of the scary angels business. He underlines the fact that the angels are jealous of his and Annabel's love, and that they are out to destroy it. Notice the way that this little subplot adds to the richness and the weirdness of the poem. With the angels in there, it isn't just a sad story about a boy and a girl. Suddenly it's a fable about a universe that is way out of whack.
And neither the angels in heaven above, Nor the demons down under the sea, (lines30-31)
Now he spreads it out a little, and adds some more characters to his creepy ghost world. Not only the jealous angels and the rich parents, but also the underwater demons are now after him and Annabel. This also makes the sea seem like kind of a scary, evil place, which is important, since it's such a major part of the poem. The spirit world is everywhere here, and we get pulled into it as we follow our speaker in his pursuit of Annabel Lee. In a way, this poem forces us to cross over into the world of death and demons along with him.
And the stars never rise, but I feel the bright eyes Of the beautiful Annabel Lee; (lines 36-37)
This is definitely a supernatural moment, even though it doesn't mention angels or ghosts or demons. Look carefully at the words the speaker uses. He doesn't say that he thinks about her when the stars rise, or that he imagines she is there. He says that he can feel her eyes on him. It's an intense, scary moment, and it plays into the sense that Annabel might not really be gone, just lurking somewhere, waiting for us to find her. This line suggests that the speaker might be a little unbalanced, but it also lets us know that he's straining to keep Annabel's memory alive.