For most of the current use cases, where NNs are used in conjunction with images, the image quality (resolution, color depth) can be low.
Consider image classification for example. The CNN extracts features from the image to tell different types of objects apart. Those features are pretty independent from the quality of the image (in reasonable bounds). Compare it with your own visual experience. Try to reduce the resolution of an image of a car step by step to figure out how little details you need until you can no longer distinguish it from a plane. This is similar to modern CNNs, which can even outperform human vision in some regards.
This changes when small details start to matter. Maybe you need to be able to detect small differences in fur patterns to tell different cat breeds apart. As soon as you lose those details, the detection rate will drop significantly.
So the answer to your question is, it depends. As long as you do not lose the important features of the image, you'll be fine with low resolution.
In case you care about the reason for the low quality of images used in machine learning - The resolution is an easy factor you can manipulate to scale the speed of your NN. Decreasing resolution will reduce the computational demands significantly.
Many CNNs even include pooling layers in their architecture, which artificially reduce the resolution further after certain processing steps. This is usually a good idea as long as you are fine with loosing positional information. You shouldn't do this when teaching the CNN to play a game, because location is highly important, but for image classification this has become an established method to increase performance.