The exploitation of bio-based foams implies an increase in the use of renewable biological resources to reduce the rapid consumption of petroleum-derived resources. Both tannins and furfuryl alcohol are derived from forestry resources and are, therefore, considered attractive precursors for the preparation of tannin–furanic foams. In addition, toughening modification of tannin–furanic foams using polyvinyl alcohol (PVOH) results in a more flexible network-like structure, which imparts excellent flexibility to the foams, whose relative properties are even close to those of polyurethane foams, which are the most used for fabrication of insoles for athletes. In addition, the addition of PVOH does not affect the thermal insulation properties of the foams by testing the thermal conductivity, resilience, and elongation at break, while reducing the brittleness of the samples and improving the mechanical properties. Also, the observation of the morphology of the foam shows that the compatibility between PVOH and tannin–furanic resin is good, and the cured foam does not show fragmentation and collapse, while the bubble pore structure is uniform. The developed flexible foam derived from biomass resources endows the foam with good thermal insulation properties and high mechanical properties, and the samples exhibit suitable physical parameters to be used as flexible insoles for athletes.