Doing yoga when you are sick?

Depends on your condition, and what yoga practice you are considering.
If you have a cold then a standard round of asanas will help clear congestion throughout the body. And pranayama is good for you in just about any circumstances.
If you are injured then you would avoid any asanas that challenge the affected area.
But in all cases listen to your body and don’t overdo it, try a limited amount and see how you feel. Some sun-salutes can loosen up the body enough to help relax and help you get to sleep.

I find the thought of someone doing yoga and having a sudden diarrhea urge quite entertaining, especially if it was a partner move.

I’d say go for it, but record it just in case.