With the increasing size of large language models, layer pruning has gained increased attention as a hardware-friendly approach for model compression. However, existing layer pruning methods struggle to simultaneously address key practical deployment challenges, including performance degradation, high training costs, and limited acceleration. To overcome these limitations, we propose E³-Pruner, a task-Effective, training-Economical and inference-Efficient layer pruning framework. E³-Pruner introduces two key innovations: (1) a differentiable mask optimization method using a Gumbel-TopK sampler, enabling efficient and precise pruning mask search; and (2) an entropy-aware adaptive knowledge distillation strategy that enhances task performance. Extensive experiments over diverse model architectures and benchmarks demonstrate the superiority of our method over state-of-the-art approaches. Notably, E³-Pruner achieves 96% accuracy, a mere 0.8% drop from the original model (96.8%) on MATH-500 when pruning 25% layers of Qwen3-32B, outperforming existing SOTA (95%), with a 1.33× inference speedup by consuming merely 0.5B tokens (0.5% of the post-training data volume).