It's not the matter of weight but mass. It is said that they don't have mass.
After all, it has energy and energy is equivalent to mass. 
I still remember that an article I read said that what I told you now is a figure of speech that physicists use to describe something about how a photon's particle-like properties are described by the language of special relativity.
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No photons don't have weight because weight is equal to product of mass and acceleration and the acceleration of photon is 0 as they move with constant speed so Photon don't have weight
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