First, as your (flat) foot comes down with each step, the foot muscles have to work harder to support the arch. As your arch flattens toward the ground, the muscles along the inner shin are pulled downward, causing strain. This can lead to shin pain, even "shinsplints."
What also happens is that as the arch flattens, the leg rotates and the knee points inward. This causes the tendons and ligaments on the inner side of the knee to stretch, while the outer part of the knee compresses.
Not surprisingly, all this can cause problems for the knee. So goes the chain reaction.
What can you do about this?
Get arch-supportive, motion-control shoes, for one. Orthotics may be necessary as well, as they'll be able to stop the arch and forefoot from flattening so much.
You might also try foot-strengthening exercises. Pushing your foot against resistance bands works well. And rest assured, very few runners with flat feet need to stop running once they're properly diagnosed and treated.