The duplicate link has good answers.
The experimental proof comes from fitting decays where photons are involved. For example the pi_0 decays into two photons. If these had a mass the energies would not balance in the fits.
Pair production would also be problematic, because a "massive photon" disappears into an electron positron pair in the field of a nucleus or an electron. Photons have not been seen to decay in vacuum.
If the photon had a mass,it would have to be twice the mass of the electron in the photon center of mass, not a negligible value , and energy and momentum balances would never work.
All these arguments are encapsulated in the theoretical framework described in the answers to the duplicate question.