If I add a CAPTCHA data element to a donation form, will it need to be entered each month in order for the credit card charges to process?
No, once the donation has passed on the initial credentials, the processor maintains the charges thereafter, so the CAPTCHA is no longer involved.
So what would happen if we unpublished the form?
Unpublished forms will also continue to process, as well as send autoresponders. Until the time frame is finished, or the user cancels the donation. The CAPTCHA is merely used to validate the form and is not actually used for the processing of the transaction itself, it is the "password" that lets you submit the form.