Receptor tyrosine kinases are single-pass membrane proteins that form dimers within the membrane. The interactions of their transmembrane domains (TMDs) play a key role in dimerization and signaling. Fibroblast growth factor receptor 3 (FGFR3) is of interest as a G380R mutation in its TMD is the underlying cause of ~99% of the cases of achondroplasia, the most common form of human dwarfism. The structural consequences of this mutation remain uncertain: the mutation shifts the position of the TMD relative to the lipid bilayer but does not alter the association free energy. We have combined coarse-grained and all-atom molecular dynamics simulations to study the dimerization of wild-type, heterodimer, and mutant FGFR3 TMDs. The simulations reveal that the helices pack together in the dimer to form a flexible interface. The primary packing mode is mediated by a Gx3G motif. There is also a secondary dimer interface that is more highly populated in heterodimer and mutant configurations that may feature in the molecular mechanism of pathology. Both coarse-grained and atomistic simulations reveal a significant shift of the G380R mutant dimer TMD relative to the bilayer to allow interactions of the arginine side chain with lipid headgroup phosphates.