Physical Danger
Working in a chicken processing plant can be plenty dangerous if you don't know what you're doing. But luckily you probably won't be down there too often.
The real danger for a CEO is stress. Long hours, travel, endless high-stakes decisions, and public outcry over chicken farm conditions are just a few of the stressors that can lead to very real health concerns. Stress can cause fatigue, headaches, heart problems, or even depression (source).
You can take steps to minimize those dangers. Plan stress-relieving activities into your day. You could do some morning meditation or afternoon exercises. And make sure you're delegating enough that you can unburden yourself of some of the more stress-inducing decisions.