There's a lot of superstition about this topic. Everyone has advice, but it's basically all based on a study of one. It pleased me no end to discover that there actually has been some research done into whether you should continue training with a cold. Some masochistic (or financially motivated?) volunteers were infected with a strain of Rhinovirus (which causes the common cold) and after they'd all succumbed to the symptoms, the ruthless scientists put them on a treadmill to find out what would happen.
There were actually two separate studies: one to determine whether performance is affected by having a cold; and one to determine whether training through a cold would affect recovery time.
The results showed that performance level (as defined by a treadmill based fitness test) is not at all affected by cold symptoms. The participants did say that their perceived level of exertion was much higher, but they were still able to run just as fast as they could before as long as they ignored the headaches, snotty noses, phlegmy throats, etc. that they were all suffering from.
The other half of the study also came out with interesting results. The participants were split into two groups: a control group and an exercising group (not quite a double blind study, but the best they could do under the circumstances). The exercising group continued running after they had been infected by the cold, while the control group were free to lie in bed and moan all day. In the end, there was no significant difference in recovery time between the two groups.
Conclusion: HTFU and keep training.
(Can't find the actual citation but there's another synopsis of the study here http://www.consorthealth.com/news-detai ... ewsID=2160
This only applies to colds though. The flu or anything more serious requires a bit of time off.
Something to be aware of for runners is the 'window of opportunity' for getting sick. After a hard/long run, your immune system is temporarily suppressed due to the stress you're placing on your body. So it's probably a good idea to stay away from sick people and crowds for a few hours after you're done running to reduce the risk of picking up something. Eating carbs straight afterwards is meant to help reduce this problem, but it's still there.