Some people will continue to work out when they don't feel well. I'll admit that I don't. For me, I have a past where I continued working out right up to the weekend I ended up being admitted to a hospital. Now, I'd ignored symptoms about a month prior, but to my own defense they had gone away. Of course, they'd just progressed to a harder to shake illness.
I'm not sure what exercise you're doing, but sometimes even when I'm not feeling enthusiastic, a walk will be okay. No jogging, not even fast walking, just a walk. I can usually be talked into a swim (but only if the weather is warm). However, there's nothing wrong with admitting that you're sick and taking it easy for a few days. Last winter I came down with a flu that was going around. Once I admitted that I was sick, I stopped working out, increased my calories and managed to shake it in half the time as most people who had it.
Now, the only danger in that is in using it as an excuse to do nothing for too long. See, not an easy answer.