Is Dataset Organization for Image Classification Necessary?
In your example, you have all the images in memory. You can simply call model.fit(trainX, trainY) to train your model. No need to organize the images in specific folder structures.
What you are referring to, is the flow_from_directory() method of the ImageDataGenerator. This is an object that will yield images from the directories, and automatically infer the labels from the folder structure. In this case, your images should be arranged in one folder per label. Since the ImageDataGenerator is a generator, you should use it in combination with model.fit_generator().
As a third option, you can write your own custom generator that yields both images and labels. This is advised in case you have a more complex label structure than one label per images; for instance in multi-label classification, object detection or semantic segmentation, where the outputs are also images. A custom generator should also be used with model.fit_generator().
I'm currently working on a program that can do binary image classification with machine learning. I have a list of labels and a list of images that i'm using as inputs which are then fed into the Inception V3 model. 
Will inputting of the dataset this way work with the inception V3 architecture? Is it necessary to organize the images with labeled folders before feeding it into the model?
Thanks for your help!