Can iCloud's Documents and Data be configured to sync on a per-app basis?

You can disable iCloud on iWork by going to the Settings app, and scroll down to the install apps settings, you will find each iWork app has it's own section, Pages, Numbers and Keynote.

In each iWork app's settings there is an option "Use iCloud". Turn this to off, and keep the Documents and Data in the iCloud menu on.

For Mac, the iWork apps do not support iCloud Document Sync. (If you enable sync on iOS, the files created on iOS will download into your library, you can't open them however, and you are unable to upload to iCloud in iWork)

In Mountain Lion they will, but it appears that iCloud Doc sync integration will be at the OS Level, rather than iWork app level. Instead being more like DropBox* where you have an area (I say area, because it's not a folder, but it's quite similar to one) for iCloud.

*Although this area only seems to appear in Open/Save Dialogs, and will only show files that are registered to the App.

The primary element of syncing is having Data to sync.

If you simply do not chose to store documents in iCloud and instead choose to store them locally on your hard drive in the way you have always previously done, then those files cannot be synchronised by iCloud, even if it remains turned "on" (Where "on" really means "available to use if you want to" rather than "It's on, you're now using it")

DayOne has a similar option, you can choose to use iCloud, or you can choose the old fashioned file/folder method. The fact that this folder can be in Dropbox and then result in subsequent synchronising is not relevant to the app.

So, in summary, using iCloud requires you to knowingly put your files into the cloud. Your existing files and indeed file locations will remain unaffected if you continue to use them.

As pointed out in another answer and subsequent edit to the question, the iWork suite '09 for Mac has yet to be updated to use iCloud. Chances are it will not be, and the OS will simply handle this for all supported applications. iCloud enabled documents in Mountain Lion simply share the same save dialogue that any normal document has, there is simply a toggle at the top of the dialogue that you can flip to say local or icloud. As such, you have the choice to save to either/or on a per document basis, and there will be no system wide setting to choose one or the other on the Mac. This mechanism will likely be brought into Lion when ML is released.

Regardless of ANY of the above, DayOne app syncing over iCloud has got absolutely no relationship with any other Application, whether is used iCloud or not, and can happily be configured to use iCloud completely independently of your choices for other apps.