We calculate the radiative lifetime and energy bandstructure of excitons in semiconducting carbon nanotubes within a tight-binding approach including the electron-hole correlations via the Bethe-Salpeter equation. In the limit of rapid interband thermalization, the radiative decay rate is maximized at intermediate temperatures and decreases at low temperature because the lowest-energy excitons are optically forbidden. The intrinsic phonons cannot scatter excitons between optically active and forbidden bands, so sample-dependent extrinsic effects that break the symmetries can play a central role. We calculate the diameter-dependent energy splittings between singlet and triplet excitons of different symmetries and the resulting dependence of radiative lifetime on temperature and tube diameter.