Safety is a paramount concern for large language models (LLMs) in open deployment, motivating the development of safeguard methods that enforce ethical and responsible use through safety alignment or guardrail mechanisms. Jailbreak attacks that exploit the \emph{false negatives} of safeguard methods have emerged as a prominent research focus in the field of LLM security. However, we found that the malicious attackers could also exploit false positives of safeguards, i.e., fooling the safeguard model to block safe content mistakenly, leading to a denial-of-service (DoS) affecting LLM users. To bridge the knowledge gap of this overlooked threat, we explore multiple attack methods that include inserting a short adversarial prompt into user prompt templates and corrupting the LLM on the server by poisoned fine-tuning. In both ways, the attack triggers safeguard rejections of user requests from the client. Our evaluation demonstrates the severity of this threat across multiple scenarios. For instance, in the scenario of white-box adversarial prompt injection, the attacker can use our optimization process to automatically generate seemingly safe adversarial prompts, approximately only 30 characters long, that universally block over 97% of user requests on Llama Guard 3. These findings reveal a new dimension in LLM safeguard evaluation -- adversarial robustness to false positives.