There is a global trend to replace the production of conventional recyclable plastics with biobased ones, allowing a sustainable alternative adapted to the current concept of a circular bioeconomy. Forest-industrial and agricultural biomass wastes (lignocellulosic biomass waste, LCBW) produce severe problems in some developing countries because they are improperly disposed of or burned in the open air. Such wastes are attractive as a raw material to produce bioplastics due to their low cost. Furthermore, low-pollution processes can complete an economical and environmentally friendly approach. This review focuses on bio-polyethylene furanoate (PEF) production from LCBW as an alternative for polyethylene terephthalate (PET), one of the most widely used fossil-based plastic. The standpoint is based on the replacement of fossil-based monomers for the manufacture of PET, terephthalic acid (TPA), and ethylene glycol by two bio-based monomers, namely 2,5-furandicarboxylic acid (FDCA) and bio-ethylene glycol (Bio-MEG). This study describes the processes to obtain each bio-monomer, as well as the resulting polymers’ performance aspects, biodegradability, environmental and economic considerations, and recycling.