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Depends on the operation there are certain tools to examine the information that is appended to the image and remove it so it can't be tracked
Also i think compressing the image to optimize website performance like instagram does might destroy the data altough i never tried but that was a great question and i will have to try to see what happens.
Image editing probably if you use paint won't make a change some other more complex programs might....

You can always set the file as read-only as to prevent any changes being made to it. If that's the concern.