There's a good article on Runner's World that talks about cardio training (especially running) in depth...here's the most pertinent quote from that article. "David Nieman, Ph.D., who heads the Human Performance Laboratory at Appalachian State University, and has run 58 marathons and ultras, uses the "neck rule." Symptoms below the neck (chest cold, bronchial infection, body ache) require time off, while symptoms above the neck (runny nose, stuffiness, sneezing) don't pose a risk to runners continuing workouts." So if you're sporting a fever or a hacking cough, it's waaaaay better to chill and recover, rather than try to run/cycle/swim through it, since doing so greatly increases your risk of future and more severe illness. If you've got a head cold, you're probably fine, but tune in to your body and watch for signs of the illness moving south. A notable exception to the "neck rule" is a sinus infection...exercising during that type of illness can lead to pneumonia or even respiratory failure. The Runner's World article suggest pool running as an alternative when you're not feeling well, since it maintains good body temperature and the moisture can ease nasal passages.
What about weight training? It's easy to understand why people are confused on this issue...it's difficult to find good answers to this question (most of the ones I came upon during my Google search were from various bodybuilding sites and not research-based at all). But I did like this one, whose advice squares with my own personal experience as well as my work with clients. The author makes a lot of good points, but the most salient one in my opinion is this: "During an infection, the body becomes catabolic (the opposite of anabolic) and breaks down muscle protein. The degree of muscle catabolism and protein loss is related to the height and duration of the fever caused by the infection...During a febrile or systemic illness, athletes should be wary of stressing muscles that are actively being broken down to fuel the body's defenses. Normal recovery after a workout can be greatly prolonged and the risk of injury may be heightened." When you're ill, your body uses energy that's normally available for exercise to fight off the illness...it also actively breaks down muscle tissue to get you back on your feet. That's the reason we feel so weak the first few days in the gym after being sick - we've literally lost muscle and have had our energy diverted to other parts of the body! So the neck rule still applies here - if the illness is systemic or below the neck, do yourself a favor and rest instead of working out. You'll get better faster and lose less muscle mass than if you're double-stressing your system!
Besides, if you cough on me and break my no-sickness streak, I'm not gonna be happy. ;) Now go drink some orange juice.
