Nowadays, a strong demand exists for printable materials with multifunctionality and proper rheological properties to overcome the limitations to deposit layer-by-layer in additive extrusion. The present study discusses rheological properties related to the microstructure of hybrid poly (lactic) acid (PLA) nanocomposites filled with graphene nanoplatelets (GNP) and multiwall carbon nanotubes (MWCNT) to produce multifunctional filament for 3D printing. The alignment and slip effects of 2D-nanoplatelets in the shear-thinning flow are compared with the strong reinforcement effects of entangled 1D-nanotubes, which govern the printability of nanocomposites at high filler contents. The mechanism of reinforcement is related to the network connectivity of nanofillers and interfacial interactions. The measured shear stress by a plate-plate rheometer of PLA, 1.5% and 9% GNP/PLA and MWCNT/PLA shows an instability for high shear rates, which is expressed as shear banding. A rheological complex model consisting of the Herschel-Bulkley model and banding stress is proposed for all considered materials. On this basis, the flow in the nozzle tube of a 3D printer is studied by a simple analytical model. The flow region is separated into three different regions in the tube, which match their boundaries. The present model gives an insight into the flow structure and better explains the reasons for printing enhancement. Experimental and modeling parameters are explored in designing printable hybrid polymer nanocomposites with added functionality.