Internet of Things (IoT) gateways integrate various sensors and compute initial decisions before transmitting data to the cloud for further processing. As the functions they need to support become increasingly complex, gateways must upgrade their hardware. Network functions (NF) and video analytics (VAs) are two typical examples of hardware requirements: NFs need specialized hardware accelerators, while VAs need parallel processing power. However, gateways are typically constrained by factors, such as power, size, and cost, leading to a need to multiplex functions and minimize hardware overprovisioning. This article proposes a novel accelerator, the transmission-analytic processing unit (TAPU), which uses multi-image FPGA to accelerate VAs and NFs for IoT gateways. We preconfigure one image for VAs and one image for NFs, then multiplex the FPGA resources in the time dimension. The TAPU system design requires both hardware and software revisions. In the hardware design, we discuss our considerations on hardware choice and present a new abstraction of hardware functions to overcome the challenge of application development on different multi-image FPGAs. For the software, we develop a fully functional TAPU system to adapt to dynamic network and VAs workloads. Our evaluation shows that TAPU utilization can reach 92%, considerably increasing VAs and network processing throughput over the current approach. We further evaluate TAPU through two case studies that support a campus traffic monitoring system and an office surveillance system, demonstrating excellent performance improvement and low overhead.