A certain "right to use" is baked into the Twitter terms and to Twitter's intended purpose of distributing content. However, pulling quotes outside of Twitter won't likely fall within that scope.
In terms of assessing a right to use a quote, it should be handled like the use of any other copyrightable work. The chief question is whether the work qualifies for copyright protection. But, assuming it does (and it can, e.g., quotes from a famous poem), andas we're considering a quote and not a larger part of the work, then the question of whether "fair use" applies becomes a legitimate consideration.
Of course, each analysis of the fair use of a quote (e.g., not receiving prior permission to use a copyrightable work) is very subjective and factor based. Fair use is usually more likely to be found when dealing with use for comment, news reporting, research, nonprofit educational use, or parody.
In other situations, particularly situations where there is a direct or indirect commercial use, then fair use probably won't apply, and you'll need permission of the author to use the quote.
Also, attribution of the source of the quote is not sufficient, by itself, to constitute permission. Attribution should be made each time a quote is used. But ultimately, you need the author's express consent.
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