Large Language Models (LLMs) have revolutionized Recommender Systems (RS) through advanced generative user modeling. However, LLM-based RS (LLM-RS) often inadvertently perpetuates bias present in the training data, leading to severe fairness issues. Addressing these fairness problems in LLM-RS faces two significant challenges. 1) Existing debiasing methods, designed for specific bias types, lack the generality to handle diverse or emerging biases in real-world applications. 2) Debiasing methods relying on retraining are computationally infeasible given the massive parameter scale of LLMs. To overcome these challenges, we propose FUDLR (Fast Unified Debiasing for LLM-RS). The core idea is to reformulate the debiasing problem as an efficient machine unlearning task with two stages. First, FUDLR identifies bias-inducing samples to unlearn through a novel bias-agnostic mask, optimized to balance fairness improvement with accuracy preservation. Its bias-agnostic design allows adaptability to various or co-existing biases simply by incorporating different fairness metrics. Second, FUDLR performs efficient debiasing by estimating and removing the influence of identified samples on model parameters. Extensive experiments demonstrate that FUDLR effectively and efficiently improves fairness while preserving recommendation accuracy, offering a practical path toward socially responsible LLM-RS. The code and data are available at https://github.com/JinLi-i/FUDLR.