Can I use Google's new CardDAV-based contacts sync to sync between my Exchange contacts and Google contacts on my iPhone?
I haven't had an opportunity to test this out, but I'm pretty sure that the iPhone silos contacts on a per-account basis. When you set up a CardDAV or Exchange account in Settings, it's unique from anything else. The contacts don't intermingle, beyond showing up in the same app.

You can think of it like having two different email accounts that your phone checks. You can access both, they'll even show up in a unified inbox, but emails in a folder on one account don't get synced to a folder on the other account.

Off the top of my head, to synchronize the two accounts, you'd probably need an app running on a computer that could synchronize between the two, and then connect your phone to either the CardDAV or Exchange server as the canonical version.