This paper reports on the 3D printing of flexible and stretchable parts based on multiwall carbon nanotube (MWCNT)/polyester-based thermoplastic polyurethane (TPU) nanocomposites. The rheological properties of the WCNT/TPU nanocomposites with different wt% of MWCNTs (0.1–3) were determined and used as guidance for the extrusion and 3D printing processes. MWCNT/TPU filaments were extruded and used for 3D printing of different flexible and stretchable parts. The mechanical, electrical, and piezoresistive response of the MWCNT/TPU nanocomposite filaments and 3D printed parts under static and monotonic loading was studied. The experimental results show that with increasing temperature and shear rate, respectively, the shear viscosity of the MWCNT/TPU nanocomposite decreases, whereas the viscosity increases with increasing wt% of MWCNTs. With the addition of MWCNTs, the elastic modulus and tensile strength of the feedstock filament all increase, enhancing the printability of TPU by increasing the buckling resistance and the stability of the 3D printed layer. The electrical conductivity of the 3D printed MWCNT/TPU nanocomposites increases with increasing wt% of MWCNTs and exceeds the conductivity of the filaments. The 3D printed MWCNT/TPU nanocomposites with 3 wt% show an electrical conductivity about 10 S/m, irrespective of the printing direction. Moreover, the 3D printed MWCNT/TPU nanocomposites exhibit good mechanical properties and high piezoresistive sensitivity with gauge factor (50–600) dependent on both strain and printing direction.