The use of alternative raw material sources in polyurethane chemistry is necessary given the limited supply of fossil fuels, their rising prices and the concern for sustainability. The production of biopolyols from edible vegetable oils such as rapeseed oil, soybean oil or sunflower oil is often proposed. In order to avoid conflict with the global food economy, non-edible or waste oils are hoped to find application in chemical synthesis. The possibility of using oils from selected fruit seeds to obtain biopolyols is analyzed in this manuscript. Five biopolyols were obtained from watermelon, cherry, black currant, grape and pomegranate fruit seeds using the transesterification reaction of the oils with triethanolamine. Thermal insulating polyurethane foams were then obtained by replacing 75% of petrochemical polyol with the biopolyols in polyurethane systems. Based on an analysis of the foaming process, it was found that the incorporation of triethanolamine molecules into the biopolyols causes a catalytic effect. The use of such biopolyols allows eliminating the catalyst from a polyurethane foam formulation. The polyurethane biofoams obtained with the pomegranate-seed-based biopolyol were characterized by the highest content of closed cells (45 vol.%). The lowest content was found for the foams containing the currant-seed-based biopolyol (9%). The foams were characterized by thermal conductivity coefficients between 32 and 35 kW/m·K and densities of approximately 40 kg/m3. Good dimensional stability and compressive strength between 100 and 250 kPa make them suitable for use in construction.