  We formulate a fracton-elasticity duality for twisted moir\'e superlattices,
taking into account that they are incommensurate crystals with dissipative
phason dynamics. From a dual tensor-gauge formulation, as compared to standard
crystals, we identify twice the number of conserved charges that describe
topological lattice defects, namely, disclinations and a new type of defect
that we dub discompressions. The key implication of these conservation laws is
that both glide and climb motions of lattice dislocations are suppressed,
indicating that dislocation networks may become exceptionally stable. Our
results also apply to other planar incommensurate crystals and quasicrystals.
