The control of heat flow in disordered materials presents a significant challenge due to the limitations of conventional phonon transport models in systems lacking periodic long-range crystal order. This study investigates energy dissipation mechanisms induced by structural irregularities, utilizing folded polymers, particularly proteins, as model systems. Proteins, macromolecules characterized by coexisting periodic amino acid chains folded into irregular three-dimensional structures, serve as useful platforms for examining the impact of irregular topologies on vibrational properties. Our research reveals an important enhancement of the phonon density of states at mid-band frequencies, diverging from the Van Hove singularities typically expected at Brillouin zone edges in perfect crystals. This state redistribution exhibits similarities to observations in some disordered electronic and optical systems, generally known as Lifshitz tails. By interpreting this effect as a resonance between multiple degrees of freedom tuned by gradients of an effective phonon confinement potential, we provide a rational for interpreting the ubiquitous "Boson Peak" reported in disordered materials. Furthermore, this study elucidates how disorder allows heat to be channeled in narrow frequency bands. To this purposes, we present mathematical tools that enable rapid and sharp estimation of the phonon density of states and thermal currents, circumventing the need for solving expensive eigenvalue problems. Our methodology may facilitate the characterization and control of heat transport in specific amorphous and disordered solids, with implications for tailoring thermal materials through strategic manipulation of structural disorder.