The SARS-CoV-2 spike protein is the first contact point between the SARS-CoV-2 virus and host cells and mediates membrane fusion. Recently, a fatty acid binding site was identified in the spike (Toelzer et al. Science 2020). The presence of linoleic acid at this site modulates binding of the spike to the human ACE2 receptor, stabilizing a locked conformation of the protein. Here, dynamical-nonequilibrium molecular dynamics simulations reveal that this fatty acid site is coupled to functionally relevant regions of the spike, some of them far from the fatty acid binding pocket. Removal of a ligand from the fatty acid binding site significantly affects the dynamics of distant, functionally important regions of the spike, including the receptor-binding motif, furin cleavage site and fusion-peptide-adjacent regions. The results also show significant differences in behaviour between clinical variants of the spike: e.g. the D614G mutation shows a significantly different conformational response for some structural motifs relevant for binding and fusion. The simulations identify structural networks through which changes at the fatty acid binding site are transmitted within the protein. These communication networks significantly involve positions that are prone to mutation, indicating that observed genetic variation in the spike may alter its response to linoleate binding and associated allosteric communication.