We investigate the effects of strain stress on the hafnium zirconium oxide ferroelectric tunnel junctions (FTJs). The impact of strain stress on each layer of the FTJ is investigated depending on the thickness of the metal capping layer and the post-metal-annealing temperature. It is revealed that, for the insulator layer, an increase in strain lead to an increased off-current in the FTJs. In contrast, increased strain stress in the ferroelectric layer directly increases the trap density, leading to an increase in the on-current of the FTJs. Furthermore, we analyze how these straininduced changes affect the performance and reliability of FTJs in neuromorphic systems. We propose optimization strategies for strain stress in FTJs based on the frequency of neural network updates, highlighting the critical balance between achieving a large dynamic range and ensuring device endurance, aligning device performance with the specific demands and conditions of neural network applications.