Gas-phase metallicity in interacting and merging galaxies offers key insights into their star formation processes and evolutionary histories. This study investigates the spatial evolution of gas-phase metallicity (i.e., oxygen abundance, 12 + log(O/H)) in these galaxies using integral field unit data from the SDSS-IV MaNGA survey, focusing on changes in metallicity gradients across different stages of interactions—from early encounters to final coalescence. By comparing interacting and merging galaxies with isolated counterparts, we identify characteristic trends in how interactions influence metallicity gradients over time. Our analysis reveals that metallicity gradients typically flatten shortly after the first pericenter passage, likely due to radial gas mixing, with later stages showing either metallicity enrichment or dilution depending on the intensity of the interaction and star formation activity. These changes can result in gradients that are either flatter or steeper than the initial profiles. Notably, we observe steeper metallicity gradients in interacting galaxies at certain merger stages, which is inconsistent with predictions from some galaxy simulations. This discrepancy emphasizes the complexity of galaxy interactions. Overall, our findings provide valuable insights into how galaxy interactions reshape metallicity distribution, enhancing our understanding of the processes driving galaxy evolution during mergers.