No to CRA, but you can inform your bank, but it's not required. Next time you file your taxes, you file with new SIN number, the months you work with old SIN also. Because in CRA system, if somebody report to your old SIN, it automatically report to the new SIN. Moreover, if you have CRA login then you need to register for CRA new login with new SIN.
Let me know if you have any questions.