The artificial intelligence-driven data deluge presents formidable challenges to conventional computing architectures, which are constrained by the von Neumann bottleneck and complementary metal-oxide-semiconductor (CMOS) scaling limits. Neuromorphic computing demonstrates breakthrough potential through its in-memory computing paradigm. Memristors, recognized as the most promising core devices in this field, excel at emulating synaptic plasticity while exhibiting nonlinear dynamic responses and conductance modulation capabilities. However, conventional inorganic memristors face critical limitations, including insufficient mechanical flexibility, biotoxicity, and non-degradability, which hinder their applications in wearable and implantable neuromorphic devices. In contrast, biomaterials and biological tissues emerge as viable platforms for overcoming these bottlenecks and realizing next-generation sustainable neuromorphic systems, owing to their exceptional biocompatibility, environmental benignity, and ultra-thin lightweight characteristics. This review focuses on cutting-edge developments in renewable biopolymer-based memristors (e.g., natural biomolecules like proteins and DNA, alongside bioengineered polymers such as poly(lactic acid) (PLA)). We elucidate how their molecular-level multifunctional groups and hierarchical structures drive high-performance memristive behaviors and bio-inspired synaptic functions. Special emphasis is placed on analyzing these devices’ superior biocompatibility and biodegradability, with in-depth discussions on how such properties enable implantable neuromorphic applications. Finally, we critically examine persisting challenges including environmental sensitivity/resistance state drift, stochastic ion migration pathways, and scalable integration hurdles. Potential countermeasures and feasible development pathways are systematically explored.