A series of poly(hexamethylene 2,5-furanodicarboxylate)-block-poly(tetrahydrofuran) (PHF-b-F-pTHF) copolymers were synthesized using a two-stage procedure, employing transesterification and polycondensation. The content of pTHF flexible segments varied from 25 to 75 wt.%. 1H nuclear magnetic resonance (NMR) and Fourier transformed infrared spectroscopy (FTIR) analyses were applied to confirm the molecular structure of the materials. Differential scanning calorimetry (DSC), dynamic mechanical measurements (DMTA), and X-ray diffraction (XRD) allowed characterizing the supramolecular structure of the synthesized copolymers. SEM analysis was applied to show the differences in the block copolymers’ morphologies concerning their chemical structure. The influence of the number of flexible segments in the copolymers on the phase transition temperatures, thermal properties, as well as the thermo-oxidative and thermal stability was analyzed. TGA analysis, along with tensile tests (static and cyclic), confirmed the utilitarian performance of the synthesized bio-based materials. It was found that an increase in the amount of pTHF caused the increase of both number-average and weight-average molecular weights and intrinsic viscosities, and at the same time causing the shift of the values of phase transition temperatures toward lower ones. Besides, PHF-b-F-pTHF containing 75 wt.% of F-pTHF units was proved to be a promising thermoplastic shape memory polymer (SMP) with a switching temperature of 20 °C.