If my ankle is stiff and sore, should I just run through the pain/discomfort to loosen it up?

Possible Problem. No. A stiff sore ankle is a sign of inflammation. This could be the result of an injury or of repeated microtrauma. You it needs to be treated with some rest, possibly nsaid's, and then a gradual return to exercise. If is does resolve with a few days of rest or it it recurs when you return to exercise - you should have it evaluated by you physician or a sports medicine physician.
Pain. Pain is a very good indicator of when to stop and rest. If the soreness an stiffness continues i would see a doctor to see why your ankle is sore and stiff.