Additive manufacturing (AM) techniques make fabricating complex designs, prototypes, and end-user products possible. Conductive polymer composites find applications in flexible electronics, sensor fabrication, and electrical circuits. In this study, thermoplastic polyurethane (TPU)-based conductive polymer composite samples were fabricated via fused filament fabrication (FFF). The effects of three important process parameters, including infill density (ID), layer thickness (LT), and fan speed (FS), on various mechanical properties (tensile and compressive properties) were investigated. It was observed that all the considered process parameters affect the mechanical properties, and they are significant parameters, as per the analysis of variance (ANOVA). From scanning electron microscopy (SEM) and optical microscopy, various combinations of parameters such as low ID, high LT, and high FS resulted in the formation of defects such as voids, cracks, and warping, which resulted in low mechanical properties. Finally, process parameter optimization was performed, resulting in a conductive polymer composite with the best possible combination of mechanical properties at high ID, low LT, and medium FS.