Whenever I'm suffering from allergies or a cold, a short bike ride or a run usually makes me feel better. I figure that just getting my blood and lungs pumping causes some of the crud to come up that usually would have sat and festered for a few days. But I'm hacking and spitting the whole time, so I wouldn't go to a gym if I were you.
Never push yourself too hard, or you'll weaken yourself when you need a strong immune system. But I think a little exercise is fine. Moving your blood around also transports antibodies quicker, right?