A lengthy Apple Discussions page shows that the problem started on the 17th of April and has persisted for 1st generation Apple TV users. Forum poster georgevargas describes the issue:
As of last night I had full access to the iTunes Store. Since this morning I was getting an Itunes Store Not Available message, and after unplugging the AppleTv the iTunes Store access completly disappeared except for movie trailers.Users have gone through the usual diagnostics, including rebooting and restoring, but have found that nothing has restored connectivity. MacRumors has confirmed the issue on our own 1st generation Apple TV.
The initial timing of the outage corresponds to the FaceTime not working for iOS 6 users, suggesting some internal changes by Apple has affected both services. Some users are speculating that internal communication upgrades related to the Heartbleed security issue could be related. Apple denied that Heartbleed affected any of its "key services" but did not elaborate on what those might be.
Apple has not provided any statement on if and when 1st generation Apple TV functionality will be restored or not. The first generation Apple TV was released in January, 2007 and was sold until September, 2010.
Update 4/22 12:00 PM PT: Apple appears to have resolved the issue preventing first-generation Apple TVs from connecting to iTunes. Many users are reporting that their Apple TVs are once again functional.
Recent Mac and iOS Blog Stories
• 'Beats Music' Updated With In-App Subscriptions, Offline Mode and More
• Facebook Updates 'Paper' App With Birthdays, Events, Photo Comments and More
• Apple Could Benefit as Samsung and Globalfoundries Standardize Mobile Chip Production
• Employees File Request to Include Steve Jobs Evidence in Anti-Poaching Lawsuit
• Former Apple CEO John Sculley: Forcing Steve Jobs Out Was a 'Mistake'
• RadioShack Drops iPhone 5s Price to $99
• Flickr Debuts Redesigned iOS App With Live Filters, HD Video Recording
• Image Editing App Pixelmator Half Off Ahead of 3.2 Update