The speaker is anxious, and the poem admits it with a strange, dry kind of honesty. The mind is "full of fears," and the speaker's first instinct is to wish it gone: "I would be glad to be rid of it." Anyone who has lain awake spinning worst-case scenarios will recognize that urge to simply switch the whole thing off.
The twist is in the middle line. "Our mouth" shifts from "my" to "our," which suggests the speaker shares a body, or a life, with someone else. Judging by the wording, the mind can't be discarded without losing the mouth, and the mouth seems to matter, perhaps for speaking, eating, or kissing. The title, "Liability," frames the mind as a burden, something that costs more than it gives. But the poem implies that what we'd like to throw away is tied to what we value.
The mood is wry, quiet, and a little sad. It's funny in a bleak way, yet the tenderness of "our" keeps it from feeling hopeless.