Residual Vector Quantization (RVQ) has become a dominant approach in neural speech and audio coding, providing high-fidelity compression. However, speech coding presents additional challenges due to real-world noise, which degrades compression efficiency. Standard codecs allocate bits uniformly, wasting bitrate on noise components that do not contribute to intelligibility. This paper introduces a Variable Bitrate RVQ (VRVQ) framework for noise-robust speech coding, dynamically adjusting bitrate per frame to optimize rate-distortion trade-offs. Unlike constant bitrate (CBR) RVQ, our method prioritizes critical speech components while suppressing residual noise. Additionally, we integrate a feature denoiser to further improve noise robustness. Experimental results show that VRVQ improves rate-distortion trade-offs over conventional methods, achieving better compression efficiency and perceptual quality in noisy conditions. Samples are available at our project page: https://yoongi43.github.io/noise_robust_vrvq/.