Charge and energy transfer in biological and synthetic organic materials are strongly influenced by the coupling of electronic states to a highly structured dissipative environment. Nonperturbative simulations of these systems require a substantial computational effort, and current methods can only be applied to large systems if environmental structures are severely coarse grained. Time evolution methods based on tensor networks are fundamentally limited by the times that can be reached due to the buildup of entanglement in time, which quickly increases the size of the tensor representation, i.e., the bond dimension. In this Letter, we introduce a dissipation-assisted matrix product factorization (DAMPF) method that combines a tensor network representation of the vibronic state within a pseudomode description of the environment where a continuous bosonic environment is mapped into a few harmonic oscillators under Lindblad damping. This framework is particularly suitable for a tensor network representation, since damping suppresses the entanglement growth among oscillators and significantly reduces the bond dimension required to achieve a desired accuracy. We show that dissipation removes the "time-wall" limitation of existing methods, enabling the long-time simulation of large vibronic systems consisting of 10-50 sites coupled to 100-1000 underdamped modes in total and for a wide range of parameter regimes. For these reasons, we believe that our formalism will facilitate the investigation of spatially extended systems with applications to quantum biology, organic photovoltaics, and quantum thermodynamics.