The growing adoption of additive manufacturing increases the need for filament materials that satisfy both technical and environmental requirements. In Fused Filament Fabrication (FFF), the binder system in metal–polymer filaments plays a key role in determining environmental impact and processing quality. This review analyses potential binder materials for FFF-based metal additive manufacturing, focusing on how their properties influence printability, thermal stability, debinding behaviour, and the mechanical performance of green parts. Comparative data on thermal and mechanical characteristics are presented for conventional and bio-based polymers. The candidate binder materials show melting points between 60 °C and 200 °C and degradation onset from 240 °C to 340 °C, defining their usability range. PLA provides the highest strength, while PCL and TPS offer flexibility and low-temperature processing. PVB/PEG ensures rigidity and clean debinding, and HPMC enables easy water removal. PLA, TPS, and HPMC exhibit the best sustainability profile due to their bio-based origin. A similar comparative analysis has not been reported in the literature.