In tire production, the molding of the tread is one of the most intricate and opaque processes. However, it also plays a significant role in tire performance. During this step, the uncured tire tread is subjected to extreme conditions and undergoes massive deformations, posing considerable challenges to simulation frameworks that could aid in the design process. For example, traditional mesh-based simulations face extreme distortion of the elements, resulting in inaccurate and unstable behavior. This study lays the foundation for a framework based on the material point method (MPM) to address these challenges. The tire tread is discretized using particles called material points. To compute their interactions, the material points are projected onto a static background grid to solve the underlying differential equations, facilitating a fast and robust computation. The focus of this contribution is on the alleviation of volumetric locking, a typical problem for nearly incompressible materials, and the interaction between the green tire and the mold, taking the unique properties of the MPM into consideration. A novel locking mitigation scheme based on the subdivision of B-splines is presented and validated. Additionally, the approach is applied to tire molding examples to investigate its efficacy.