The answer to this question is that theoretically, they cannot be zeros; but experimentally, the mass of the photon is practically zero while its energy is the product of Planck's constant and its frequency. The mass of the neutrinos (electron, muon or tau) can all be practically zero but theoretically, they also cannot be zero.

The relativistic mass of a fermion (e.g. electron) is the product of its rest mass and the gamma factor (inverse the square root of one minus the ratio of the square of fermion speed over the speed of light). This formula only make sense if and only if the speed of the fermion never become exactly zero.