I have a checkbook from my US-based online bank, but the address on the checks is now out of date. What is the best way to destroy the checkbook? Do I need to notify my bank that I am not using the checks anymore?
To get rid of old checks, just use a shredder or other supervised destruction method. You generally aren't liable if someone steals your checks and uses them fraudulently; however, you don't want to give someone the opportunity to do so, either.
No, you don't need to notify your bank. As long as you have new checks that have your current address, you are fine. The bank doesn't care if the numbers on the checks they receive are consecutive or not.
I burn them, myself. Shredders are easy to overcome with modern technology unless they randomly destroy the material as well.
That said, I don't use them anymore. Too much liability for not enough return these days. Better to use bill pay for such things so that it is drawn on an internal account to the bank.