Frequency Modulated Continuous Wave (FMCW) radars can measure subtle chest wall oscillations to enable non-contact heartbeat sensing. However, traditional radar-based heartbeat sensing methods face performance degradation due to noise. Learning-based radar methods achieve better noise robustness but require costly labeled signals for supervised training. To overcome these limitations, we propose the first unsupervised framework for radar-based heartbeat sensing via Augmented Pseudo-Label and Noise Contrast (Radar-APLANC). We propose to use both the heartbeat range and noise range within the radar range matrix to construct the positive and negative samples, respectively, for improved noise robustness. Our Noise-Contrastive Triplet (NCT) loss only utilizes positive samples, negative samples, and pseudo-label signals generated by the traditional radar method, thereby avoiding dependence on expensive ground-truth physiological signals. We further design a pseudo-label augmentation approach featuring adaptive noise-aware label selection to improve pseudo-label signal quality. Extensive experiments on the Equipleth dataset and our collected radar dataset demonstrate that our unsupervised method achieves performance comparable to state-of-the-art supervised methods. Our code, dataset, and supplementary materials can be accessed from https://github.com/RadarHRSensing/Radar-APLANC.