The resonance scattering of Ly alpha photons with neutral hydrogen atoms in the intergalactic medium not only couples the spin temperature to the kinetic temperature but also leads to a heating of the gas. We investigate the impact of this heating on the average brightness temperature of the 21-cm signal from the Cosmic Dawn in the context of the claimed detection by the EDGES low-band experiment. We model the evolution of the global signal taking into account the Ly alpha coupling and heating and a cooling which can be stronger than the Hubble cooling. Using the claimed detection of a strong absorption signal at z approximate to 17 as a constraint, we find that a strong Ly alpha background is ruled out. Instead the results favour a weak Ly alpha background combined with an excess cooling mechanism which is substantially stronger than previously considered. We also show that the cooling mechanism driven by the interaction between millicharged baryons and dark matter particles no longer provides a viable explanation for the EDGES result when Ly alpha heating is taken into account.