I haven’t received a direct response to this forum post (obviously) or my email inquiries. At least, Twitter’s response wasn’t clear to me: “Using Mechanical Turk to distribute the text part of the Tweet to be hand labelled should be a problem.” I couldn’t tell if that was a typo and should say “shouldn’t be a problem?”
Anyway, what I believe would work, and not violate the terms of service is to host HITs myself. For example, take a twitter ID, display it on a web page (conforming to the Twitter policies), and let the Turk user rate or classify. There seem to be some libraries set up for the hosting (not necessarily the twitter part): http://gureckislab.org/mtworkshop/#2, https://github.com/NYUCCL/psiTurk
I thought I would post a follow up if anyone else is interested. I DO NOT have this confirmed from Twitter, but it seems like it would stay in the bounds of their policies.