The role played by environment in galaxy evolution is a topic of ongoing debate among astronomers. There has been little success in elucidating the degree to which environment can alter, re-shape, or drive galaxy evolution, that is, using either observations or simulations. However, our knowledge of the effect of environment on gas metallicity gradients remains limited. Here we present our analysis of the gas metallicity gradients for a sample of ten Fornax cluster galaxies observed with MUSE as part of the Fornax3D project. We used detailed maps of emission lines to determine precise values of gas metallicity and metallicity gradients. The integrated gas metallicity of our Fornax cluster galaxies shows slightly higher metallicities (∼0.045 dex) in comparison to a control sample. In addition, we find signs of a mass and metallicity segregation from the center to the outskirts of the cluster. By comparing our Fornax cluster metallicity gradients with a control sample we find a general median offset of ∼0.04 dex/Re, with eight of our galaxies showing flatter or more positive gradients. The intermediate infallers in our Fornax sample show more positive gradients with respect to the control sample. We find no systematic difference between the gradients of recent and intermediate infallers when considering the projected distance of each galaxy to the cluster center. To identify the origin of the observed offset in the metallicity gradients, we performed a similar analysis with data from the TNG50 simulation. We identify 12 subhalos in Fornax-like clusters and compared their metallicity gradients with a control sample of field subhalos. This exercise also shows a flattening in the metallicity gradients for galaxies in Fornax-like halos, with a median offset of ∼0.05 dex/Re. We also analyzed the merger history, Mach numbers (ℳ), and ram pressure stripping of our TNG50 sample. We conclude that the observed flattening in metallicity gradients is likely due to a combination of galaxies traveling at supersonic velocities (ℳ > 1), which are experiencing high ram pressure stripping and flybys.