  We calculate the spectrum of interlayer neutral excitons in
transition-metal-dichalcogenide WeSe$_2$/MoSe$_2$ heterobilayers in the
$R$-stacking configuration. Most saliently, we show that, similarly to neutral
excitons and trions in monolayer transition-metal dichalcogenides, the spectrum
is sensitive to the Berry curvature and thus quantum-geometric effects
underlying the electron and hole wave functions. Due to the spatial separation
between the electron and hole constituting the exciton in different layers, the
Berry-curvature-induced splitting of the between the $2p_+$ and the $2p_-$
exciton states is smaller than for monolayer excitons. Furthermore, we
investigate the dependence of the exciton spectra on the dielectric environment
and the twist angle between the two layers. Finally, the long-lived moir\'e
interlayer exciton ground state ($1s$) enhances the possibility of creating
brightened $2p_{\pm}$ states using a circularly polarized medium-infrared probe
from the $1s$ ground state. As a result, we determine the polarizability of the
$1s-2p_{\pm}$ transition, following by two-level dressed model for the optical
Stark effect.
