Deep neural networks (DNNs) have been increasingly explored for receiver design because they can handle complex environments without relying on explicit channel models. Nevertheless, because communication channels change rapidly, their distributions can shift over time, often making periodic retraining necessary. This paper proposes a zero-overhead online and continual learning framework for orthogonal frequency-division multiplexing (OFDM) neural receivers that directly detect the soft bits of received signals. Unlike conventional fine-tuning methods that rely on dedicated training intervals or full resource grids, our approach leverages existing demodulation reference signals (DMRS) to simultaneously enable signal demodulation and model adaptation. We introduce three pilot designs: fully randomized, hybrid, and additional pilots that flexibly support joint demodulation and learning. To accommodate these pilot designs, we develop two receiver architectures: (i) a parallel design that separates inference and fine-tuning for uninterrupted operation, and (ii) a forward-pass reusing design that reduces computational complexity. Simulation results show that the proposed method effectively tracks both slow and fast channel distribution variations without additional overhead, service interruption, or catastrophic performance degradation under distribution shift.