My appointment with CERN comes to an end shortly. I have accepted an offer by a private think-tank to deal with issues similar to those that I am currently working on at CERN. Are there any precautions that I need to take?

Discuss the situation with your hierarchy.  You should not work on any task that could create a conflict of interest, or the appearance thereof, between CERN and your future employer.
Bear in mind that your obligation not to disclose any confidential information you have acquired in connection with your work at CERN continues to exist after you leave the Organization.