Their sources have indicated that the program won't technically be shut down, but Amazon won't be releasing any updates to the Kindle Developer Kit, the suite of tools and documentation that app developers can use to create apps for the Kindle ereaders. Amazon also won't be accepting new app submissions, though given their past behavior I'm not sure that developers will be able to tell the difference.
Update: I have heard back from one developer who says this is news to him. His relationship with Amazon is as good as ever, and that leads me to think that Amazon is more likely banishing certain developers but not completely shutting down the platform.
I have heard complaints that Amazon's approval process for Kindle Active Content is so capricious they make Apple look sane and reasonable. Amazon has rejected apps for what the developers describe as ridiculous reasons, blocked app updates with no explanation, and generally made life difficult for the developers.
I've also been told off the record that Amazon plays favorites with developers. Amazon has reportedly neglected to approve an app from a developer so a similar app from a preferred developer can be released first.
In any case, now that a developer has called this story into question all I can say is that this situation is far from clear.
There's no sign of any change to the relevant Developer Portal on Amazon.com, though I will note that the page is still described as being in beta. That is a surprise given that Kindle Active Content was originally launched in early 2010.
If I receive confirmation of this story, I will update this post.