There are 8 small bones and 5 complex ligament structures in the wrist. A traditional push up position compresses the wrist bones and strains the ligaments because of the acute angle the position requires. Doing ten push ups in this position might be okay, but trying to do, for example, a 6 minute body weight routine from this position will lead to wrist pain, fatigue, and possibly injuries.
Of course, the stress on the bones and ligaments is intensified by the weight of the athlete. In addition, the motion of pushing up and down during exercise can actually add to the strain.