A proper remedy for the overexploitation of biomass and biobased materials in the bioeconomy is the valorization of biorefineries’ side streams into meaningful products. Hence, in pursuit of a cascade utilization of renewables, a unique biorefinery byproduct was investigated for its biogas potential, specifically methane, in continuously operated anaerobic filters. For this purpose, 5-Hydroxymethylfurfural process-wastewater, after supplementation of necessary nutrients, was diluted down to 10, 20, 30, 40, and 50 gCOD/L concentrations and thereafter tested individually at 43 °C and 55 °C. Maximum methane conversion efficiency at either temperature was observed for test substrates with 10 gCOD/L and 20 gCOD/L concentrations. At 43 °C, the anaerobic filters exhibited their highest biogas yields when supplied with the 30 gCOD/L feedstock. Further exposure of the mesophilic and thermophilic consortia to the ensuing 5-Hydroxymethylfurfural process-wastewater dilutions compromised the stability of the anaerobic process due to the soaring concentrations of short-chained volatile fatty acids. The supplementation of necessary nutrients to unlock the methane potential of the given recalcitrant substrate appears insufficient. Techniques like micro aeration, photolysis, or the use of activated carbon in the fixed bed might have the ability to enhance the biochemical methane conversion of such feedstock; otherwise, the introduction of trace elements alone may be adequate if aiming for platforms (volatile fatty acids) via anaerobic technologies.