Is it a problem of bal tashchit to throw out edible leftovers if one prefers to do so for no good reason?
The general prohibition against needless destruction, derived from the verse on fruit trees, concerns not destroying directly or indirectly anything that may be of use to people.
The Talmudic sage Rabbi Yishmael makes another logical inference: if the Torah warns us not to destroy fruit trees, then we should be even more careful about not destroying the fruit itself. Currently, in Israel, Rabbi Moshe Yitzhak Forehand notes that all rabbinic authorities agree, based on this teaching, that it is forbidden from the Torah to destroy edible fruit. This applies to all food that is fit to be eaten, and not only the fruit of trees.
Rabbi Moshe Aaron Poleyeff (1888–1967, Europe and U.S.) wrote that overeating may be a double transgression of bal tashchit—first by wasting food, and second by harming one’s body.Rabbi Shaya Karlinsky (contemporary, Israel) considers taking more food than one can eat at a buffet to be bal tashchit.
So it seems that (in answer to the text of the question) one may not throw out edible leftovers.
The title of the question is “do I have to eat the leftovers”. Whether one should eat the leftovers will depend on whether overeating is involved.