Much of the answer on this question is dependent on the specifics of copyright transfer and the specific policies of the journal. Here is a journal article about copyright and author self-archiving. Among authors, the topic of copyright transfer is not very well-understood.
What consequences are typically enforced:
According to this article by chroniclevitae.com , the typical consequences for authors who publish their site to a social media site (Academia.edu in this example) is a notice to the violating website to remove the copyright'd content, However, even this action is somewhat limited, as academia.edu received a peak of 2,800 requests over a few week span for a site that hosts over 2.3 million journal articles.
Remember: As Bledsoe points out, this is still murky territory. Publishers, journals, and scholars are all feeling their way around.
Possible extent of consequences:
When an author agrees to publication from a publisher, the publisher receives not only the words and graphics, but also (sometimes) the copyright from the author. If the publication required a copyright transfer, a copyright violation is being committed by uploading a published paper, even by the author of the paper. The possible consequences are as diverse as typical copyright violations, but can be up to and including paying money to the publisher for loss of revenue.
Money damages in copyright infringement actions are commonly awarded under three legal theories, actual damages, profits, and statutory damages.
Source for additional reading and understanding of copyright infringement and the possible monetary consequences from a legal point-of-view.