An Apple support article acknowledges a bug that affects playing ringtones in the latest (9.0.3) version of iTunes. The symptom is that, if you attempt to play ringtones from iTunes on your Mac, the ringtones will not play. Instead, an Authorize Computer dialog appears, informing you that you are not authorized to play the ringtone and prompting you to enter the Apple ID and password for your iTunes account.
Presumably you are already authorized, despite what the dialog says. You can confirm this by entering the requested data and clicking the Authorize button. At this point, a dialog should pop up that states: "This computer is already authorized."
Unfortunately, none of this has any affect on your ability to play the ringtone. If you try again, the same Authorize Computer dialog re-appears, leaving you in a seemingly endless loop. There is no work-around offered at this time. This bug applies only to ringtones purchased from the iTunes Store.
The good news is that you should still be able to sync these ringtones to your iPhone, where they will play correctly. If it's important that you hear the ringtones in iTunes on your Mac, you'll likely have to wait for the next iTunes update.
In the unlikely event that a ringtone was purchased via a different account than your own, see this Apple support article for general troubleshooting advice regarding authorizing computers in iTunes.
This story, "Bugs & Fixes: Ringtones Won't Play in ITunes" was originally published by Macworld.