Contact-rich tasks pose significant challenges for robotic systems due to inherent uncertainty, complex dynamics, and the high risk of damage during interaction. Recent advances in learning-based control have shown great potential in enabling robots to acquire and generalize complex manipulation skills in such environments, but ensuring safety, both during exploration and execution, remains a critical bottleneck for reliable real-world deployment. This survey provides a comprehensive overview of safe learning-based methods for robot contact-rich tasks. We categorize existing approaches into two main domains: safe exploration and safe execution. We review key techniques, including constrained reinforcement learning, risk-sensitive optimization, uncertainty-aware modeling, control barrier functions, and model predictive safety shields, and highlight how these methods incorporate prior knowledge, task structure, and online adaptation to balance safety and efficiency. A particular emphasis of this survey is on how these safe learning principles extend to and interact with emerging robotic foundation models, especially vision-language models (VLMs) and vision-language-action models (VLAs), which unify perception, language, and control for contact-rich manipulation. We discuss both the new safety opportunities enabled by VLM/VLA-based methods, such as language-level specification of constraints and multimodal grounding of safety signals, and the amplified risks and evaluation challenges they introduce. Finally, we outline current limitations and promising future directions toward deploying reliable, safety-aligned, and foundation-model-enabled robots in complex contact-rich environments. More details and materials are available at our \href{ https://github.com/jack-sherman01/Awesome-Learning4Safe-Contact-rich-tasks}{Project GitHub Repository}.