Is it possible to disable Copy & Paste on Linux so that people using Putty or any other ssh client can't copy & paste from the server to their own machine?
You wouldn't be able to disable the functionality of another machine's rendering of the output your server sends to it.
The output on putty uses the rules setup in Putty for copying the data displayed on the screen. Meaning that the data displayed on the screen is the output from your server to the terminal and the source (Linux) no longer has control of that data.
You could change the installed options for each of the terminals connecting to your server assuming you have admin power over those machines as well.
As a suggestion for controlling things on the server side I would restrict the ability to display data that you wouldn't want copy/pasted or restrict access to the server at all from ssh for users. Meaning you could wrap a lot of functions in sudo or not allow access to databases or portions of the file system.