There are many personalities that use Twitter to promote themselves, and are active on the site.
If I sell a product that quotes or reproduces these tweets is it fair use? Let's assume that we are talking about tweets of 140 chars or less.
The length of a literary work doesn't determine if its use is fair use.
Copyright does not cover names, titles of works, catchwords/catchphrases/advertising slogans etc. or lists of ingredients (like in a recipe or chemical instructions), however, the procedure can be copyright.
The work must also be original. If the tweet copies something that already exists then there is no copyright in the tweet - if the thing that is copied has copyright protection then the tweet itself may be a violation.
That said, assuming that the Tweet enjoys copyright protection, the owner will be the author: presumably the owner of the Twitter account. For anyone else to legally use it, the use must be either:
For Item 1, the Twitter terms of service say:
You retain your rights to any Content you submit, post or display on or through the Services. By submitting, posting or displaying Content on or through the Services, you grant us a worldwide, non-exclusive, royalty-free license (with the right to sublicense) to use, copy, reproduce, process, adapt, modify, publish, transmit, display and distribute such Content in any and all media or distribution methods (now known or later developed).
In case you don't know what that means, Twitter gives you this plain English summary:
Tip: This license is you authorizing us to make your Tweets on the Twitter Services available to the rest of the world and to let others do the same.
So, you can get permission from the copyright holder or from Twitter. Well, Twitter has given permission:
Except as permitted through the Twitter Services, these Terms, or the terms provided on dev.twitter.com, you have to use the Twitter API if you want to reproduce, modify, create derivative works, distribute, sell, transfer, publicly display, publicly perform, transmit, or otherwise use the Twitter Services or Content on the Twitter Services.
Providing you use their API, you can "reproduce, modify, create derivative works, distribute, sell, transfer, publicly display, publicly perform, transmit, or otherwise use the ... Content ..."
Therefore, fair use/dealing considerations are irrelevant.