Just like anything else dealing with emotions, keeping them out of your startup is a sure way to put a bullet in it.
Emotions like joy, trust, interest and surprise are critical to engagement with customers. Marketing is basically raising interest. Product deals with the rest. The CEO broadcasts all of them, if he's good at his job.
You also can't keep negative emotions from injecting themselves at the most inopportune time. How you handle those negative emotions, on the other hand, has far reaching implications.