Do I classify all types if they are mutually exclusive I am trying to classify an image that can represents 3 states. Up, down or Middle.

If the image does NOT represent Up or Down, then it is by default Middle.

Should I train my CNN with a dataset including all three, or just Up and Down?

Which would this make classification more or less accurate?
You won't be able to train your model correctly if you set only two classes. Because for all the samples the model will only predict Up or Down. This will cause even the Middle classes to be predicted as Up or Down, which will reduce your model's accuracy. Hence you need to keep all three classes as labels.