Additive manufacturing (AM) has arisen as a transformative technology for manufacturing complex geometries with enhanced mechanical properties, particularly in the realm of continuous fiber-reinforced polymer composites (CFRPCs). Among various AM techniques, fused deposition modeling (FDM) stands out as a promising method for the fabrication of CFRPCs due to its versatility, ease of use, flexibility, and cost-effectiveness. Several research papers on the AM of CFRPs via FDM were summarized and therefore this review paper provides a critical examination of the process-printing parameters influencing the AM process, with a focus on their impact on mechanical properties. This review covers details of factors such as fiber orientation, layer thickness, nozzle diameter, fiber volume fraction, printing temperature, and infill design, extracted from the existing literature. Through a visual representation of the process parameters (printing and material) and properties (mechanical, physical, and thermal), this paper aims to separate out the optimal processing parameters that have been inferred from various research studies. Furthermore, this analysis critically evaluates the current state-of-the-art research, highlighting advancements, applications, filament production methods, challenges, and opportunities for further development in this field. In comparison to short fibers, continuous fiber filaments can render better strength; however, delamination issues persist. Various parameters affect the printing process differently, resulting in several limitations that need to be addressed. Signifying the relationship between printing parameters and mechanical properties is vital for optimizing CFRPC fabrication via FDM, enabling the realization of lightweight, high-strength components for various industrial applications.