Dynamic neural radiance fields (dynamic NeRFs) have demonstrated impressive results in novel view synthesis on 3D dynamic scenes. However, they often require complete video sequences for training followed by novel view synthesis, which is similar to playing back the recording of a dynamic 3D scene. In contrast, we propose OD-NeRF to efficiently train and render dynamic NeRFs on-the-fly which instead is capable of streaming the dynamic scene. When training on-the-fly, the training frames become available sequentially and the model is trained and rendered frame-by-frame. The key challenge of efficient on-the-fly training is how to utilize the radiance field estimated from the previous frames effectively. To tackle this challenge, we propose: 1) a NeRF model conditioned on the multi-view projected colors to implicitly track correspondence between the current and previous frames, and 2) a transition and update algorithm that leverages the occupancy grid from the last frame to sample efficiently at the current frame. Our algorithm can achieve an interactive speed of 6FPS training and rendering on synthetic dynamic scenes on-the-fly, and a significant speed-up compared to the state-of-the-art on real-world dynamic scenes.