Just to follow up on what @James said ("Hiding the button will not keep them from accomplishing the task in other ways"), is there a reason why you are trying to use technology to stop people from doing this, instead of just explaining to them why they cannot abuse the copyright...?
As James notes, if you remove the button, people can still find ways to download any content that they are viewing on the screen (everything online is just a screenshot away from being downloaded...), and if these people are your TAs, people whom you have put in a position of trust, then surely you can explain to them that one of the things you trust they will do is to respect copyright, and to contact you if they have a real reason for needing to download copyright materials in order to find a workable solution...?
Admittedly, sometimes even faculty are not clear about copyright, fair use, etc., and the same would be true of TAs, but that seems to me to make it a teachable moment, rather than trying to police it with technology.
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