Edge computing is a distributed computing paradigm that collects and processes data at or near the source of data generation. The on-device learning at edge relies on device-to-device wireless communication to facilitate real-time data sharing and collaborative decision-making among multiple devices. This significantly improves the adaptability of the edge computing system to the changing environments. However, as the scale of the edge computing system is getting larger, communication among devices is becoming the bottleneck because of the limited bandwidth of wireless communication leads to large data transfer latency. To reduce the amount of device-to-device data transmission and accelerate on-device learning, in this paper, we propose Residual-INR, a fog computing-based communication-efficient on-device learning framework by utilizing implicit neural representation (INR) to compress images/videos into neural network weights. Residual-INR enhances data transfer efficiency by collecting JPEG images from edge devices, compressing them into INR format at the fog node, and redistributing them for on-device learning. By using a smaller INR for full image encoding and a separate object INR for high-quality object region reconstruction through residual encoding, our technique can reduce the encoding redundancy while maintaining the object quality. Residual-INR is a promising solution for edge on-device learning because it reduces data transmission by up to 5.16 x across a network of 10 edge devices. It also facilitates CPU-free accelerated on-device learning, achieving up to 2.9 x speedup without sacrificing accuracy. Our code is available at: https://github.com/sharc-lab/Residual-INR.