Photos aren't often printed on foam board, but they are mounted on them.

One of the major commandments when producing works of art on a rigid substrate like wood, masonite or heavy board is "whatsoever thou doest to one side, that shalt thou also do unto the other side". Essentially, that means making sure that however one modifies the stresses upon or the absorbency of one side of the ground, you have to match on the other side. In the case of mounting photos to foam boards, you are covering the paper on one side of the board with glue and another piece of paper having at least a sizing and probably a glossier ink-receptive coating to one side of the board. That will significantly affect the rate at which moisture migrates into the paper component on that side of the board, while the other side is free to absorb humidity and expand. So it isn't the printed side contracting that's the problem, it's the back expanding.

You can mount another piece of the same type of paper on the back of the board, making sure that any varnishes, etc., used on the front are also used on the back. (Commonly, one would use a lighter/cheaper weight of paper on the back -- it's the glue and coatings that matter more than the paper itself.) Or you could step up to a more rigid support (Gatorfoam, painter's panel with bracing, gessoed hardboard).

Mounting on foam board is, though, at best a short-term display solution. It isn't archival, and isn't meant to be. It isn't a substitute for proper framing.