Elastomers with distinctive viscoelastic behavior are widely utilized in vibration and noise reduction. However, most elastomers are derived from petroleum-based sources and their functionality are limited to a single purpose. In this work, a biobased polyurethane (bio-PU) elastomer was synthesized from biobased poly(trimethylene ether) glycol, and then a H-bond cross-linking strategy was implemented by incorporating biobased tannic acid (bio-TA) to fabricate biobased PU/TA (bio-PU/TA) damping composites. The structures and performances of the fabricated composites were systematically characterized. The results illustrated that the strong H-bond interactions between bio-PU and bio-TA molecular chains greatly enhanced the mechanical and damping performances of the composites, in which the tensile strength, elongation at break, and tan δ max of bio-PU/TA composites were enhanced by 173.5, 68.7, and 55.4%, respectively. Moreover, the introduction of bio-TA with abundant polyphenol structures remarkably improves the capacity for residual carbon formation and the ultraviolet (UV)-shielding of bio-PU, leading to an upgraded bio-PU/TA composite flame retardancy and enhanced UV-shielding properties in the 280−400 nm UV region. The fabricated bio-PU/TA composites offer a promising strategy for developing damping materials with multifunctional performances.