yeah that is one of the most annoying myths that keeps getting spread around. while 'rabbit starvation' (a colloquial term for protein poisoning; it can also be called 'mal de caribou' or caribou sickness,) IS real, it only occurs when you eat only protein, no fat or carbs or anything else, for an extremely long time - like months to years. it gets that name because it's something that did happen historically with some frequency in people who were consuming nothing but wild rabbits or other lean game over long winters.
domestic rabbits are fat. they have been bred to gain weight in both muscle and fat, live sedentary lifestyles, and eat a diet formulated for production. and even if wild rabbit is your only protein source forever, the average modern person with access to a grocery store will not be eating ONLY rabbit for every meal, nothing else. eat a balanced diet with both fat and carbs in addition to protein and you'll be fine, i promise.