In many situations, excitons—bound particle-hole pairs above an insulating ground state—carry an electric dipole moment, allowing them to be manipulated via coupling to an electric field. For two-dimensional systems, we demonstrate that this property of an exciton is uniquely determined by the quantum geometry of its eigenstates, and demonstrate its intimate connection with a quantity which we call the quantum geometric dipole. We demonstrate that this quantity arises naturally in the semiclassical equations of motion of an exciton in an electric field, adding a term additional to the anomalous velocity coming from Berry's curvature. In a uniform electric field, this contributes a drift velocity to the exciton akin to that expected for excitons in crossed electric and magnetic fields, even in the absence of a real magnetic field. We compute the quantities relevant to semiclassical exciton dynamics for several interesting examples of bilayer systems with weak interlayer tunneling and Fermi energy in a gap, where the exciton may be sensibly described as a two-body problem. These quantities include the exciton dispersion, its quantum geometric dipole, and its Berry's curvature. For a simple example of two gapped-graphene layers in a vanishing magnetic field, we demonstrate that there is a nonvanishing quantum geometric dipole when the layers are different, e.g., have different gaps, but vanishes when the layers are identical. We further analyze examples in the presence of magnetic fields, allowing us to examine cases involving graphene, in which a gap is opened by Landau level splitting. Heterostructures involving transition metal dichalcogenides materials are also considered. In each case, the quantum geometric dipole and Berry's curvatures play out in different ways. In some cases, the lowest energy exciton state is found to reside at finite momentum, with interesting possible consequences for Bose condensation in these systems. Additionally, we find situations in which the quantum geometric dipole increases monotonically with exciton momentum, suggesting that the quantum geometry can be exploited to produce photocurrents from initially bound excitons with electric fields, without the need to overcome an effective barrier via tunneling or thermal excitation. We speculate on further possible effects of the semiclassical dynamics in geometries where the constituent layers are subject to the same or different electric fields.