How do we identify the fact that the photon has no mass? In Newton's book "Opticks", he proposed the particle theory of light to demonstrate the refraction phenomena. In his theory, he postulated that the tiny particles of light have mass and experience an attractive force from air to denser media (like water). Using the postulate, he derived the Snell's law and predicted the speed of light in water should be faster.
Now we know the speed of light in the vacuum is absolute and the mass of the photon is zero. 
But how do we identify the fact that the photon has no mass, experimentally or theoretically? Or is there any experiment to verify the fact just like measure the speed of light?
 A: There are two key data here:


*

*The measured invariance of the speed of light with respect to boosts, e.g. by the Michelson-Morley experiment, shows that the speed of light must be very close to the universal signal speed limit derived from an Ignatowskian approach to special relativity. See my answer here for further information. Thus, the rest mass $m_0$ term in the relativistic expression for the photon's total energy $E^2 - p^2c^2 = m_0^2\,c^4$ must be very small. Note that modern versions of the Michelson-Morley experiment (using resonant interferometers rather like small versions of the LIGO ones) show that the change in the speed of light $\Delta c_L$ wrought by the addition of the Earth's motion is of the is of the order of $\Delta c_L/c_L<10^{-15}$. Another less accurate, but everyday and perhaps more intuitively compelling, observation along these lines is that if you push, say electrons, harder and harder to higher and higher energies of motion in a particle accelerator, one can easily observe their speed to asymptote to a speed very near to that of light. Of course, this experiment is measuring the universal signal speed limit $c$, rather than the speed of light $c_L$, so this measurement together with one of light speed show that $c_L\approx c$ experimentally. Again, any evidence that the speed of light is near to the universal signal speed limit is evidence for small photon mass.

*The more accurate limits come from electromagnetic observations. Maxwell's equations would take a well-known form - the Proca equation - if the photon had nonzero rest mass. In turn, a nonzero rest mass implies some striking, readily measured experimental results, such as the existence of nonzero electric fields inside closed conductors, which are not observed, and also astronomical observations of galactic plasma. See my answer here as well as the section   "Experimental checks on photon mass" on the Wikipedia  "Photon" page for more details. The photon mass bound gotten by these methods is $10^{-14}\mathrm{eV}/c^2$, or about $1.6\times10^{-50}\mathrm{kg}$, i.e. about $10^{-20}$ electron masses.
A: Experiments don't determine exact quantities because of small errors inherent in making measurements. So, there can be an experimental upper limit on the photon rest mass, but not experimental zero, because that would imply infinite precision, and our technology can only show so many digits after the decimal point.
https://www.princeton.edu/~romalis/PHYS312/Coulomb%20Ref/TuCoulomb.pdf
A: 
But how do we identify the fact that: the photon has no mass, experimentally or theoretically? Or is there any experiment to verify the fact just like measure the speed of light?

The short answer is that nothing that has rest mass can travel at the speed of light, as it would have infinite kinetic energy, see https://en.wikipedia.org/wiki/Kinetic_energy#Relativistic_kinetic_energy_of_rigid_bodies.
However, a thought experiment suggests that if you contain photons in a box that is perfectly reflective in its interior, the box will have inertia due to the photons, which goes a long way towards proving that photons kind of do have mass.
In the end, it depends what you mean by mass. In Newtonian mechanics, mass is the thing that appears in $F_{net}=m\cdot a$. Beyond Newtonian mechanics, mass that appears in different equations is not always the same concept (if you wanted to be super-clear, you could give each a different name and then the question "do photons have Mass$_A$ or Mass$_B$ or both" could be answered precisely).
A: There exists an enormous amount of data in high energy physics  which has been derived using as a fact, an axiom, that the photon, and  a few other  particles , have zero mass. These are encapsulated in the standard model of particle physics having used the special relativity equations maximally. 
The continuous validation of the standard model  is also an experimental proof that its basic premises, including special relativity and the zero mass particles, are valid.
