The geometry of electronic bands in a solid can drastically alter single-particle charge and spin transport. We show here that collective optical excitations arising from Coulomb interactions also exhibit unique signatures of Berry curvature and quantum geometric tensor. A nonzero Berry curvature mixes and lifts the degeneracy of l≠0 states, leading to a time-reversal-symmetric analog of the orbital Zeeman effect. The quantum geometric tensor, on the other hand, leads to l-dependent shifts of exciton states that is analogous to the Lamb shift. Our results provide an explanation for the nonhydrogenic exciton spectrum recently calculated for transition-metal dichalcogenides. Numerically, we find a Berry curvature induced splitting of ∼10 meV between the 2px±i2py states of WSe2.