My body "says" to me "you better lie down", "you can't do anything".  But the question is whether I am reading the signals correctly.

Diane, an ironman triathlon athlete, starts to get the flu and she goes out and "runs it off", she says. However, that is contraindicated; see below.

I, on the other hand, fold my tent and just lie there in misery, thinking maybe I should still be able to get things done. 

Yes, the body does need rest to heal, so that is wise to rest fully.

When the body is sick it has "toxins" of some sort in it.  It would make sense to drink lots of water and perhaps take some vitamins to help flush out the bad stuff through diluting it all and letting it flow.

But what about exercising when that is the last thing I feel like doing?

Unfortunately, in a sense, it is appropriate that you move the body so that the lymph system can move the lymph - no movement makes it very slow - and thus recovery is very slow.  Exercise at least with walking or the equivalent 5 minutes at least 4 times a day.  Movement also has the side benefit of making one feel better.

Basic, simple stretching that is non-strenuous is excellent, as it moves the lymph system very well.

When you have a fever, you are supposed to rest, drink liquids and wait for your body's immune system to beat the germs. Exercising with a fever will only make you sicker or keep your sicker longer.   

In any event, pay attention to what your body is telling you.  And... easy does it!

Sleep, sleep, sleep, sleep, rest, rest, rest... give the body the time to do the repair process.

And don't eat a bunch of ice cream or other crappola - it'll delay the healing process.

In any event, look up the instructions for the illness and/or symptoms you are experiencing and then follow them.  Use WebMD.com or even eHow.com.