The response of individual species to climate change may alter the composition and dynamics of communities. Here, we show that the impacts of environmental change on communities can depend on the nature of the interspecific interactions: mutualistic communities typically respond differently than commensalistic or parasitic communities. We model and analyse the geographic range shifting of metapopulations of two interacting species - a host and an obligate species. Different types of interspecific interactions are implemented by modifying local extinction rates according to the presence/absence of the other species. We distinguish and compare three fundamentally different community types: mutualism, commensalism and parasitism. We find that community dynamics during geographic range shifting critically depends on the type of interspecific interactions. Parasitic interactions exacerbate the negative effect of environmental change whereas mutualistic interactions only partly compensate it. Commensalistic interactions exhibit an intermediate response. Based on these model outcomes, we predict that parasitic species interactions may be more vulnerable to geographic range shifting than commensalistic or mutualistic ones. However, we observe that when climate stabilises following a period of change, the rate of community recovery is largely independent of the type of interspecific interactions. These results emphasize that communities respond delicately to environmental change, and that local interspecific interactions can affect range shifting communities at large spatial scales.