We propose the existence of a symmetry-protected topological Dirac nodal line (DNL) magnonic phase in layered honeycomb collinear antiferromagnets even in the presence of spin-orbit Dzyaloshinskii-Moriya interaction. We show that the magnon spin Nernst effect, predicted to occur in strictly two-dimensional (2D) honeycomb collinear antiferromagnets cancels out in the layered honeycomb collinear antiferromagnets. In other words, the magnon spin Nernst effect in each 2D antiferromagnetic layer cancels out the succeeding layer. Hence, the Berry curvature vanish in the entire Brillouin zone due to the combination of time-reversal and space-inversion (PT ) symmetry. However, upon symmetry breaking by an external magnetic field, we show that a non-vanishing Berry curvature and Chern number protected topological magnon bands are induced in the noncollinear spin structure. This leads to an experimentally accessible magnon thermal Hall effect in the PT symmetry-broken topological DNL magnonic phase of layered honeycomb antiferromagnets. We propose that the current predicted results can be experimentally investigated in the layered honeycomb antiferromagnets CaMn2Sb2, BaNi2V2O8, and Bi3Mn4O12(NO3).