Despite recent advances in dexterous manipulations, the manipulation of articulated objects and generalization across different categories remain significant challenges. To address these issues, we introduce DART, a novel framework that enhances a diffusion-based policy with affordance learning and linear temporal logic (LTL) representations to improve the learning efficiency and generalizability of articulated dexterous manipulation. Specifically, DART leverages LTL to understand task semantics and affordance learning to identify optimal interaction points. The {diffusion-based policy} then generalizes these interactions across various categories. Additionally, we exploit an optimization method based on interaction data to refine actions, overcoming the limitations of traditional diffusion policies that typically rely on offline reinforcement learning or learning from demonstrations. Experimental results demonstrate that DART outperforms most existing methods in manipulation ability, generalization performance, transfer reasoning, and robustness. For more information, visit our project website at: https://sites.google.com/view/dart0257/.