Multi-task semantic communication (SC) can reduce the computational resources in wireless systems since retraining is not required when switching between tasks. However, existing approaches typically rely on task-specific embeddings to identify the intended task, necessitating retraining the entire model when given a new task. Consequently, this drives the need for a multi-task SC system that can handle new tasks without additional training, known as zero-shot learning. Inspired by the superior zero-shot capabilities of large language models (LLMs), we leverage pre-trained instruction-tuned LLMs, referred to as fine-tuned language net (FLAN), to improve the generalization capability. We incorporate a mixture-of-experts (MoE) architecture in the FLAN model and propose MoE-FLAN-SC architecture for multi-task SC systems. Our proposed MoE-FLAN-SC architecture can further improve the performance of FLAN-T5 model without increasing the computational cost. Moreover, we design a multi-task feature extraction module (FEM) which can adaptively extract relevant features across various tasks given the provided features and signal-to-noise ratio (SNR). Simulation results show that our proposed MoE-FLAN-SC architecture outperforms three state-of-the-art models in terms of the average accuracy on four different unseen tasks.