Because the auxiliary verb does governs the syntax, and does wants a non-finite form (moan, shake, wait), whereas is gone is finite. It could be "...then go?"
That statement about the dove, horse, and lizard is analogous to
Why does he ask then runs? ungrammatical
Grammatical would be
Why does he ask then run?
In contemporary standard English, does ... be is a no-go, so that "does ... be gone" is not a viable option:
Why does the dove moan or the horse shake its mane? Or the lizard wait
on the white wall then be gone? non-standard
does...be is archaic in standard English, and is used now only in some (usually regional) dialects