Well, that would be straight-up tortious interference. But I feel like now that clients have moved on to higher-level protocols, and exist on so many platforms, Twitter would have a hard time blocking use of API keys extracted from the official apps. If I have time, I'll retrieve them myself...
Technically, if those keys could be extracted (I'm no expert on that), that would be the easier option, as Twitter cannot really afford to revoke those without massive headaches on their end. You likely couldn't distribute those keys, but I have no idea if automatic extraction at install time would be fair game on a large scale.
Actually, keys are not copyrightable, so the only obstacle I know of to distributing them is the DMCA - which I think doesn't apply here, since the key does not circumvent an access or copy control measure (it doesn't bypass the requirement to log in or anything like that, and Twitter.app has no technical measures to prevent you from copying tweets).
My bad with the copyright, although it looks like Twitter leaves themselves nice leeway to kill at least your personal account for it, as it is forbidden to "access or search..the Services by any means other than through our currently available, published interfaces that are provided by Twitter (and only pursuant to those terms and conditions)"[1], which I would read to exclude the (slightly-modified, I believe) access Twitter's own apps get.
