Due to the current trends in sustainable development and the reduction in the use of fossil fuels (Green Deal strategy and the circular economy), and thus, the increased interest of the polyurethane industry in polyols derived from renewable sources, it is important to study the impact of these polyols on the flammability of new bioelastomers. The goal of this study was to check the influence of biobased polyols, such as tall oil (TO)-based polyols, soybean oil (SO)-based polyol, and rapeseed oil (RO)-based polyol, on the reduction in the burning and fume emissions of polyurethane and poly(urea)urethane elastomers (EPURs and EPUURs). The thermal stability of these materials was tested using thermogravimetric analysis (TGA). In turn, the flame retardancy and smoke emissions were checked using a cone calorimetry test. The released gases were identified using TGA coupled with Fourier transform infrared (FT-IR) spectroscopy (TGA/FT-IR). Moreover, the morphological and structural characteristics of the char residues were characterized using FT-IR and scanning electron microscopy (SEM) with energy-dispersive spectroscopy (EDS). The obtained data were compared to the results received for elastomers produced with petroleum substrates. The addition of biobased polyols led to a reduction in the burning as a result of the formation of char, especially RO polyol. Moreover, the TO and RO polyols increased the thermal stability of the elastomers.