We propose a task-oriented multiuser wireless communication framework for distributed classification based on a MAP-driven system design under wireless channel impairments. By deriving tractable approximations of the MAP error bound, the proposed approach enables the design of learning-based feature extraction and precoding strategies. Unlike existing approaches that optimize intermediate reconstruction, information-theoretic, or feature-separability objectives, the proposed formulation directly optimizes objectives derived from the MAP decision error, thereby providing a more direct connection to the final classification performance. Simulation results demonstrate that the proposed schemes achieve higher classification accuracy than their counterparts, with lower or comparable computational complexity.