Does a "Photon Box" have gravitational mass?

Two points:

• even massless particles (such as photons) "fall in a gravitational field";
• the amount of mass actually doesn't change how it falls, either in Newtonian gravitation (where gravitational and inertial masses coincide), or in General Relativity (where free objects follow space-time geodesics), provided the box mass is small enough not to change the gravitational force/curvature.

But, rigorously, any amount of energy in the box will influence the space-time curvature. What brings us to a possibly more interesting question: "would objects with mass fall towards the box?". And the answer is: "yes". A box with enough light inside would create the same gravity well as, e.g., the Earth.

It would also have weight. Due to gravitational red-shift the momentum of a photon hitting the bottom is higher than the same photon hitting the ceiling of the box. If you calculate this I think you end up with a 'mass' of 1/3 E/c^2.

The box will indeed fall, for the same reason that light is lensed by the gravitational field of the Sun (for example). The gravitational field curves spacetime, and the photons will follow the curved paths. The photons will also in general exert their own gravitational field, though a very negligible one in all realistic scenarios, and the details are a bit complicated.