We report on linewidth analysis of optical transitions in InAs/GaAs coupled quantum dots as a function of bias voltage, temperature, and tunnel coupling strength. A significant line broadening up to 100 $mu$eV is observed at hole tunneling resonances where the coherent tunnel coupling between spatially direct and indirect exciton states is maximized, corresponding to a phonon-assisted transition rate of 150 ns${}^{-1}$ at 20 K. With increasing temperature, the linewidth shows broadening characteristic of single-phonon transitions. The linewidth as a function of tunnel coupling strength tracks the theoretical prediction of linewidth broadening due to phonon-assisted transitions, and is maximized with an energy splitting between the two exciton branches of 0.8$-$0.9 meV. This report highlights the linewidth broadening mechanisms and fundamental aspects of the interaction between these systems and the local environment.