Narrative theology is a system of interpreting the Bible subjectively, based on the perspective (opinion) of a group or individual reader. The Scriptures, therefore, will have different meanings manufactured by different readers according to their own individual perspectives, biases, and prejudices.
By its own reckoning, narrative theology does not set any objective meanings to the Bible or any part thereof. This means, for example, that if feminists interpret the Bible a certain way and liberation theologians another, there is no way to call any of them "wrong."
History
Holman's Dictionary states the following regarding some of the history of narrative theology.
"Black preachers became masters at "telling the story." They practiced narrative theology long before it was suddenly "discovered" by some seminaries in the 1970's.
"Even though the congregations had heard these Bible stories, they were anxious to hear them again. Not only did the repetition reinforce the story, but each man gave the stories new vitality through his interpretation." (Holman Bible Dictionary article entitled Black People and Biblical Perspectives).
Begging Questions
Narrative theology begs several important questions. Perhaps the most important question is if there is no absolute truth or meaning to the Biblical text, then why should a person care about the arguments made by those who advocate this form of theology. Their perspective is just as subjective, and therefore, worthless, as anybody else's.
If there is no objectivity in interpreting any text in the Bible, then (say) a sign saying, "Danger High Voltage," might mean that a French gourmet restaurant is nearby, and not that a hazardous amount of electricity might electrocute the careless.
If authorial intent does not matter at all and is indeterminable objectively, then anything indeed goes with Narrative theology (and a host of liberal sacred cows is slain with the general epistemological wreckage coming from deconstructionist literary theory).
Therefore, if there is an objective interpretation of (say) a biology textbook, there can also be one (in principle) of Scripture using the grammatico-historical method of interpretation and objectively based hermeneutics.