Poly(3-hydroxybutyrate) (PHB) is a biodegradable polymer, whose applicability is limited by its relatively poor mechanical properties and narrow processing window. In this paper, different natural phenol-based additives, including tannic acid, grape bagasse extract, and lignin were used as thermal and processing stabilizers for PHB. The thermal stability of both neat and doped PHB samples was studied by rheology and calorimetry. The experimental results showed that neat PHB massively degrades and that the addition of phenol additives enhances the thermal stability of PHB, thus widening the processing window of the polymer. Physical and chemical interactions between polymer and additive were considered as key factors to interpret the experimental data. The additives affected the melt crystallization kinetics of PHB inhibiting crystallization upon cooling. Moreover, the grape bagasse extract in PHB was found to be effective to improve processability and photooxidative stability of the biopolymer. Lignin promoted thermal stabilization of the polyester matrix under temperature conditions close to those of processing, although tensile tests showed that heterogeneity in filler morphology can affect mechanical properties of the polymer. Molecular weight measurements on thermally degraded samples also suggest that cross-linking may occur at high additives’ content. The described results are of interest for the development of sustainable alternatives to synthetic polymer additives, by increasing the applicability of bio-based materials.