What part of Apple's review process might lead to them reject apps with not-complete features? Will Apple decline an app from the App Store if a test feature is in place? I've looked over the generic App review guidelines but don't know how or where the line is drawn.

For an example specific to my case, we want to add a button that will redirect a user to a coming soon page on our website.  We ultimately want to see how many users are clicking on the button to gauge how much interest there is in this product/feature.
Yes.  This has happened to me.  Apple sends your submission through a light testing regime by a human.  If they pick up an egregious bug or substantially incomplete feature, they may reject your submission.

You could change the verbiage on the button to make it clear the feature is not yet implemented, which would probably pass.

One suggestion is to have a live blog and link to discussions, using a redirect from your web server to track which button was pressed in the app.  This way, reviewers arrive at a logical page while you could track which version of the app was used. Why risk rejection for submitting something with an incomplete feature when you can have a simple "coming soon" link before you submit your app for review?