No need to call ASPCA—these burning dogs are not your fluffy best friends. At first they look rather hellish, with rolling eyeballs and burning legs. And since they come up from the basement (some kind of underworld), we know they can't be bringing good news. But they're also a big part of the speaker's joyous transformation after eating all that poetry. So there's something of a silver lining in their presence, especially when he himself essentially turns into one of them and licks the librarian's hand. Yum.
Line 9: The dogs look pretty ominous here as they come up from the basement. Dogs stuck in basements aren't the happiest pups in the world. And since they "come up" after the poems are gone, we get the sense that they represent some of the speaker's darker thoughts and feelings.
Lines 10-11: Since their eyeballs are rolling and their legs are burning, they're looking pretty hellish at this point. They don't strike us as too friendly.
Lines 16-18: But even though they look hellish, they still have a pretty significant influence on our speaker, who's now snarling, barking, and romping with joy. So it's as if they're part of him now, or as if he's joined their pack. If they represent some of the speaker's darker thoughts, it makes sense that they'd be part of him. If we see this as him joining their pack, it could mean that the poetry has transformed him into a someone (or something) more in touch with his animalistic, primal nature. (Cue the wolf howls.)