Yes, a consistent yoga practice can help you increase flexibility and strengthen your core. No, you do not need to lose fat before starting a yoga practice. I would suggest that you begin by attending classes (as opposed to doing yoga with a DVD), because a teacher can help you modify any poses that you find difficult, until you become more flexible. If attending classes sounds a little intimidating, look for a teacher that will schedule a private session with you, and help you come up with a routine that you can do on your own at home.
My yoga teachers have said it’s important to ‘start where you are’ and move forward from there!