Multi-task learning (MTL) is a standard learning paradigm in machine learning. The central idea of MTL is to capture the shared knowledge among multiple tasks for mitigating the problem of data sparsity where the annotated samples for each task are quite limited. Recent studies indicate that graph multi-task learning (GMTL) yields the promising improvement over previous MTL methods. GMTL represents tasks on a task relation graph, and further leverages graph neural networks (GNNs) to learn complex task relationships. Although GMTL achieves the better performance, the construction of task relation graph heavily depends on simple heuristic tricks, which results in the existence of spurious task correlations and the absence of true edges between tasks with strong connections. This problem largely limits the effectiveness of GMTL. To this end, we propose the Generative Causality-driven Network (GCNet), a novel framework that progressively learns the causal structure between tasks to discover which tasks are beneficial to be jointly trained for improving generalization ability and model robustness. To be specific, in the feature space, GCNet first introduces a feature-level generator to generate the structure prior for reducing learning difficulty. Afterwards, GCNet develops a output-level generator which is parameterized as a new causal energy-based model (EBM) to refine the learned structure prior in the output space driven by causality. Benefiting from our proposed causal framework, we theoretically derive an intervention contrastive estimation for training this causal EBM efficiently. Experiments are conducted on multiple synthetic and real-world datasets. Extensive empirical results and model analyses demonstrate the superior performance of GCNet over several competitive MTL baselines.