El Capitan server repeatedly asks client machines for Calendar and Contacts passwords

I have a similar problem.

To add to this comment, my iPhone and iPad appear to be updating the calendar just fine, the problem seems to be limited to the El Capitan Contacts App vs my OS X Server, with no problems seen vs the iCloud address book.

I'm seeing the problem from 3 different accounts on 2 different MBP15s

I'm having the same issue with a couple OS X El Capitan 10.11.2 client machines. Its asking for both Calendar and Contacts server password every few minutes.

I've looked through the Server.app logs and Console.app on the clients machine but haven't found any helpful information. I think its an overlooked bug, that we can't really do anything about. I just hope they fix this soon unlike the iTunes (DRM) HDCP issues I've been having since Mavericks.

This sounds suspiciously like Mavericks Server Keychain not properly storing information network users. A problem that, you can tell from the title, has been around for some time. But the issue is for Network users and user homes not local users. If using Network users and user homes read below.

For Network users: Check the keychains of the affected users in User's home / Library / Keychains. You will find a keychain folder for each client computer the user works on. If you find items in the keychain folders with "corrupt" in the name you are having the same problem as the above discussion. If that is indeed the problem, a quick work around is to throw out the keychain folders containing the corrupt files. Reboot the machine in use and log back in. The user has a fresh keychain for the machine they are on and will have to reenter all of their saved passwords. Then always reboot the machine instead of logging out. Rebooting solves the problem but is not very convenient. If you got this far, look into a scripted solution in the middle of page 13 in the discussion above.