1. To walk carefully, so as not to disturb what is underfoot or nearby. This is rare alpine vegetation, so tread lightly. Tread lightly so you don't wake up the baby.
2. To be extra tactful in one's dealing with someone so as not to offend or aggravate. The boss is really irritable today, so if you have bad news for him, tread lightly. Tread lightly if you ask Mrs. Smith for extra credit—she usually gets annoyed with those requests.