We study interactions between localized scatterers on metallic carbon nanotubes by a mapping onto a one-dimensional Casimir problem. Backscattering of electrons between localized scattering potentials mediates long-range forces between them. We model spatially localized scatterers by local and nonlocal potentials and treat simultaneously the effects of intravalley and intervalley backscattering. We find that the long-range forces between scatterers exhibit the universal power-law decay of the Casimir force in one dimension, with prefactors that control the sign and strength of the interaction. These prefactors are nonuniversal and depend on the symmetry and degree of localization of the scattering potentials. We find that local potentials inevitably lead to a coupled valley scattering problem, though by contrast nonlocal potentials lead to two decoupled single-valley problems in a physically realized regime. The Casimir effect due to two-valley scattering potentials is characterized by the appearance of spatially periodic modulations of the force.