Four short lines, and they seem to describe a relationship that is both thrilling and dangerous. The title, "Alewife," points toward a woman who serves ale, and the poem plays on that idea. The first pair of lines sets love up as a "private war," something fought between two people where no one else can see. Then comes the image of "dwelling in fire," which suggests living inside constant heat. It could mean passion, but also pain, since nobody can settle comfortably in flames.
The second pair turns toward intimacy. The speaker has kissed these lips, and the comparison to ale hints that the kisses are intoxicating, perhaps something to be sampled and craved again. The phrasing is a little teasing, so it's not entirely clear how far the comparison goes. Ale can warm you, but it can also leave you dizzy.
The mood is playful but edgy: desire mixed with conflict, and a speaker who seems both drawn in and slightly wary. The poem leaves much unsaid, and that gap is part of its charm.