Laser sails propelledby gigawatt-scale ground-based laser arrayshave the potential to reach relativistic speeds, traversing the solarsystem in hours and reaching nearby stars in years. Here, we describethe danger interplanetary dust poses to the survival of a laser sailduring its acceleration phase. We show through multiphysics simulationshow localized heating from a single optically absorbing dust particleon the sail can initiate a thermal runaway process that rapidly spreadsand destroys the entire sail. We explore potential mitigation strategies,including increasing the in-plane thermal conductivity of the sailto reduce the peak temperature at hot spots and isolating the absorptiveregions of the sail that can burn away individually.