This review article aims to summarize the potential of using renewable natural resources, such as lignin and tannin, in the preparation of NIPUs for wood adhesives. Polyurethanes (PUs) are extremely versatile polymeric materials, which have been widely used in numerous applications, e.g., packaging, footwear, construction, the automotive industry, the lighting industry, insulation panels, bedding, furniture, metallurgy, sealants, coatings, foams, and wood adhesives. The isocyanate-based PUs exhibit strong adhesion properties, excellent flexibility, and durability, but they lack renewability. Therefore, this study focused on the development of non-isocyanate polyurethane lignin and tannin resins for wood adhesives. PUs are commercially synthesized using polyols and polyisocyanates. Isocyanates are toxic, costly, and not renewable; thus, a search of suitable alternatives in the synthesis of polyurethane resins is needed. The reaction with diamine compounds could result in NIPUs based on lignin and tannin. The research on bio-based components for PU synthesis confirmed that they have good characteristics as an alternative for the petroleum-based adhesives. The advantages of improved strength, low curing temperatures, shorter pressing times, and isocyanate-free properties were demonstrated by lignin- and tannin-based NIPUs. The elimination of isocyanate, associated with environmental and human health hazards, NIPU synthesis, and its properties and applications, including wood adhesives, are reported comprehensively in this paper. The future perspectives of NIPUs’ production and application were also outlined.