Today, the answer is indeed no. As Google says in the AdSense Terms of Service, explaining why you might not be getting ads on a protected page:
"Your web pages are behind a login: At this time, pages that require a login can not be easily visited by our crawlers. Additionally, since these types of pages are not easily visited for approval and it is difficult for our AdSense specialists to verify that pages behind a log in follow Google AdSense policies, we are unable to offer support for these pages."
However, as of early November, it's clear that the AdSense team is testing out just this capability, as this notice the AdSense team sent to a colleague explains:
"You are one of a select group of publishers that has been chosen to try out our new Site Authentication feature. Site Authentication allows publishers to give our crawler access to login-protected content so that users can receive targeted ads. While there is no obligation for you to try out this new feature, you may find it helpful if you have pages of your site protected behind a login screen.
"As an example, an online newspaper site may protect content behind a login for premium readers. Without Site Authentication, the publisher can't show readers targeted ads on those premium pages because our crawler has not been able to access the content. Using Site Authentication, the publisher can now specify details for our crawler to pass the login screen and reach the protected content, allowing AdSense to serve relevant ads to the readers."
As you can read, the program's called Site Authentication, or, perhaps, AdSense Site Authentication and it'll offer exactly what you -- and many other people -- seek in terms of additional functionality with AdSense. If you didn't receive this message and are part of the AdSense program, I'd shoot 'em an email asking if you can be part of the beta test. It can't hurt!
And if you haven't yet signed up for AdSense, I encourage you to start here: Get started with Google AdSense.
Good luck! Come back in a few months and let us know how it's working out, please.
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As blogging software continues to integrate better with forums, I think membership or subscriber only content will become much more common.