The tactile Internet (TI) has emerged as an advanced communication technology that enables interaction between humans and machines. However, TI use cases demand low latency, high reliability, and high-capacity requirements to achieve high quality of experience (QoE) in their applications. IoT networks provide transmission services to heterogeneous applications, and the inclusion of high-demand Quality of Service (QoS) requirements by TI applications will increase the difficulty of guaranteeing end-to-end QoS. We proposed a fine-grained traffic differentiation in tactile Internet scenarios using the Recursive Inter-Networking Architecture (RINA) QoS cubes and the Quantitative Timeliness Agreement Multiplexor (QTA-Mux) flow prioritization scheme. We designed four RINA QoS cubes to meet the delay and packet loss requirements of TI, Augmented Reality/Virtual Reality (AR/VR), and IoT applications and to provide fine-grained traffic differentiation. Also, we designed the QTA-Mux policy to enable a trade-off between delay and packet loss and adequately scheduling the service data unit (SDU) for delivering. We implemented a RINA network to test the proposed schemes in IoT environments and conducted several experiments to evaluate the delay and packet loss. The results showed that RINA-based prioritization and traffic differentiation proposed guaranteed end-to-end QoS to support TI applications. Thus, the next generation of IoT networks can use RINA to guarantee the QoS requirements of TI applications.