The actual entity that's protesting here is "Academia.edu" - their place in the ecosystem is rather unclear and I would say tenuous. If you look at Elselvier's copyright blurb, as an author with them you are allowed to email a PDF of the finished paper to anyone who asks you for one, and you can do what you like with the manuscript that you submit to them before peer review - what you're asked not to do is to put the full text that's undergone peer review, has likely been improved by copy editing, and by incorporating referee comments received during peer review, and by being typeset - into a third party website (such as Academia.edu).
Elselvier and publishers like them do add some finite, measurable, non-zero value to a paper - they filter out some errors via peer review, by rejecting nonsense, and they improve other papers that are reviewed, corrected and resubmitted. They also typeset the manuscript. This takes a bit of time for the editor involved and it's not unreasonable that they should get a few hundred dollars for their time.
Academia.edu is jumping in after the work of reviewing the paper has been performed - then it's only the cost of indexing and hosting the PDF - which is truly negligible. Why should Academia.edu be allowed to host papers, when they have not been involved in reviewing or improving the text? Reading the linked post, the take-down notice is being served on the third party, Academia.edu, not the author.
The author of the paper does have the rights to distribute the paper to anyone who asks for a copy via email, and the RCUK is moving academics in the UK towards a scheme where papers that are not open access cannot be used to secure government funding from April 2013. Elsevier will have to adapt to that or it will be left behind. There's fairly strict rules, such as the paper must be guaranteed to be available for 10 years from the publication date, with the expiry clock reset after each access to the URL.
What we will need is the equivalent of a public right-of-way someone to download each link once every 10 years, in the same way that public footpaths are kept open by committed ramblers walking across the public rights of way every 7 years or so.