Oil in a recipe acts as a lubricator for a batter (or a flavour carrier) and it helps to keep the cake moist. Oil has a high density that will help the food remain firm at room temperature, needed for dense baking items such as a zucchini or carrot cake and some muffins.

In general it is not a good idea to substitute ingredients in a baking recipe. However, it is possible with some quick bread recipes to replace some of the oil with a fruit purée such as apple sauce. If you want to experiment with this, substitute about 3/4 cups of applesauce for every 1 cup of oil. In addition to this add 2 T of oil as a vehicle to distribute flavour in the cake. If your recipe calls for all-purpose flour I would also recommend that you switch to cake flour and make sure you do not over bake or your cake will be dry.
 
Give it a try and see how it works,
Christine