Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot

We give two infinite families of examples of closed, orientable, irreducible 3-manifolds M such that b₁(M) = 1 and π1(M) has weight 1, but M is not the result of Dehn surgery along a knot in the 3-sphere. This answers a question of Aschenbrenner, Friedl, and Wilton and provides the first examples of irreducible manifolds with b₁ = 1 that are known not to be surgery on a knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while each member of the other family is not even homology cobordant to any Seifert fibered 3-manifold. None of our examples are homology cobordant to any manifold obtained by Dehn surgery along a knot in the 3-sphere.