In multi-dimensional classification (MDC), each instance is associated with labels from multiple potentially interdependent class dimensions. However, existing approaches often overlook the fact that different semantic dimensions may require distinct feature representations. Additionally, irrelevant and redundant features in the feature space can adversely affect model performance. To address these issues, a feature selection approach based on evolutionary multi-tasking named Fest is proposed for MDC. It treats feature selection for each class dimension as a separate subtask for evolution, ensuring the selected features effectively capture the semantics of each dimension. To effectively identify and select shared features between correlated class dimensions, Fest introduces an exploration mechanism for feature interaction that considers class dependencies. Extensive experiments are conducted on eleven benchmark datasets as well as on four state-of-the-art MDC approaches. Experimental results clearly demonstrate that selecting dimension-specific features instead of all features can significantly improve the classification performance of existing MDC approaches.