Disable Saving/Exporting Results of TSQL Query in SSMS? I really don't want to go in to too much detail as to WHY I want to do this. Just know that it would be ideal for my use case. 

Is there a way to disable a specific database role from using the Copy, Copy with Headers, Save Results As..., Print... functionality in the results window in SSMS?
No, there is no way to handle this from the database/instance, unless you just deny them access to the database at all.

SSMS is a client-side tool, and the things that it can do with the data you've already let it consume are outside of server-side controls. Even if you could take those menu items away, there is always the PrintScreen button.