Zero surgery on a Seifert fiber space

What you call a 0-surgery on a Seifert manifold is a move that typically does not produce a Seifert manifold. It is a move that "kills the fiber" and it produces a graph manifold, homeomorphic to a connected sum of lens spaces.

Since this may be a potential source of confusion, let me mention that a a 0-surgery on the $(p,q)$-torus knot is not a "0-surgery" in the above sense, and it produces a Seifert manifold that fibers over the orbifold $(S^2, p, q, pq)$ with Euler number zero (since $H_1$ is infinite cyclic).