Fused filament fabrication (FFF) stands out as one of the most widely used additive manufacturing (AM) techniques, attracting considerable attention in recent years. Despite the remarkable strides witnessed in FFF-based AM technology, challenges persist in fabricating robust, high-performance, and functional components for demanding real-world applications. In comparison to polymer/continuous carbon fiber-reinforced polymer (CCFRP) composites produced using conventional processing methods, those fabricated via FFF technology often exhibit inherent weaknesses in their mechanical properties, including reduced strength and anisotropic behavior. These deficiencies stem from inherent imperfections such as voids and poor interfacial adhesion, a consequence of the layer-by-layer deposition nature coupled with the dual-phase composition of composite materials. While traditional remedies such as process refinement, in situ processing, and post-processing techniques have been pursued, there remain limitations in their effectiveness. This paper begins with an overview of the categories of voids and interfaces encountered in FFF-fabricated polymer/CCFRP composites. Subsequently, the existing improvement strategies are reviewed, and the focus shifts to an in-depth exploration of material modification approaches. This encompasses both matrix material modification and carbon fiber (CF) surface enhancements, examining their influence on void formation, interface quality, and overall performance of printed parts. Finally, the prospects for future research directions in material modification of three-dimensional (3D) printed polymers and CCFRP composites are highlighted.