Must one obey a father who forbids his daughter from posting her picture on Snapchat, knowing that the account is for girls only, and is his prohibition out of fear of the evil eye or in pursuit of his whims?

It is obligatory for you to think well of your father and obey him in what he commands, so long as it does not involve disobedience to Allah Almighty. His forbidding you from using Snapchat or other social media, out of fear of the dangers that might result from their use, stems from his compassion and concern for your well-being.