The flow of polymer solutions past solid obstacles or through porous media gives rise to rich physical phenomena over a wide range of spatial and temporal scales. Viscoelasticity, in particular, can induce a strong nonlinear response with an increase of flow resistance even for a solution whose viscosity decreases in simple shear flow. Various hypotheses have been proposed to explain this phenomenon but a clear picture of the pore-scale mechanisms involved and their impact upon larger scales is still lacking. Here, we show that localized zones of large polymeric stress, known as birefringent strands, drive the flow of an Oldroyd-B fluid through two-dimensional arrays of cylinders. Combining a recently developed numerical scheme with high performance computing, we find that these strands generate a complete reorganization of the flow with an increase of stagnation zones, a reinforcement of preferential paths and a splitting of flow channels. Furthermore, we show that this reorganization is the source of an increase in the viscous dissipation of the solvent and also that the stretching of polymer molecules in the strands is associated with entropy production. Both these phenomena yield a global increase in dissipation that can be directly linked to the increase of flow resistance. Our results demonstrate that the birefringent strands-not the elongational viscosity-drive the flow of viscoelastic fluids through porous media and that the increase of flow resistance can occur even at steady state, before the transition to elastic turbulence.