There is an example above with mortality for people with/without heart disease based on eating hamburgers; if you partition by having heart disease, you falsely would infer that in some of the population, hamburgers aren't bad for you. Obviously, the real issue is that actually the hamburgers are what initially caused the heart disease.

I am sure there are better examples, but this one will suffice. If anyone gives it some though and comes up with a better one, that would be great.