Additive manufacturing (AM) has attracted huge attention for manufacturing metals, ceramics, highly filled composites, or virgin polymers. Of all the AM methods, material extrusion (MEX) stands out as one of the most widely employed AM methods on a global scale, specifically when dealing with thermoplastic polymers and composites, as this technique requires a very low initial investment and usage simplicity. This review extensively addresses the latest advancements in the field of MEX of feedstock made of polymers highly filled with metal particles. After developing a 3D model, the polymeric binder is removed from the 3D-printed component in a process called debinding. Furthermore, sintering is conducted at a temperature below the melting temperature of the metallic powder to obtain the fully densified solid component. The stages of MEX-based processing, which comprise the choice of powder, development of binder system, compounding, 3D printing, and post-treatment, i.e., debinding and sintering, are discussed. It is shown that both 3D printing and post-processing parameters are interconnected and interdependent factors, concurring in determining the resulting mechanical properties of the sintered metal. In particular, the polymeric binder, along with its removal, results to be one of the most critical factors in the success of the entire process. The mechanical properties of sintered components produced through MEX are generally inferior, compared with traditional techniques, as final MEX products are more porous.