It's a nice sentiment, but thanks to the Open Access movement, most recently published papers are already available online somewhere, and the people talking about #pdftribute are the people most likely to have already shared their papers. Important historical papers, even those that are now public domain, are still behind paywalls. As Aaron said, "even under the best scenarios, their work will only apply to things published in the future. Everything up until now will have been lost." 
Sharing your own papers is nice, but it's also safe. It's not really challenging the status quo.
Edit: I overstated my point. I don't know that "most" papers are already available; it certainly varies a lot by field. I kind of doubt that many #pdftributers are people that weren't previously sharing their papers, though.
That is not true. I work in neuroscience and often find myself not having access to "just released" papers. Some of them may appear as preprints months later, if the author bothers to do so. I 've tried to email authors for copies a few times with no response. Even my institution doesn't provide access to all the journals we need (probably the situation is different in the US). There are actually underground websites for searching paywalled journals through proxies.
TBH, historic papers are not that interesting anymore, since the most important ones are cited in more recent research. It's the cutting edge research where it's more annoying. What's more important though is that closed-access is depriving science of the ability to use automated tools for textual analysis. I hope this unfortunate event will motivate more people to realize that having unrestricted access to scientific results is an extremely important issue.
I am in astronomy and it is common practice to post papers on arxiv.org. Because of the need for journal subscriptions and the time it takes for papers to be published, this has become the primary way we keep up with the field. I would say this has revolutionized math and the physical sciences.
Perhaps one of the reasons for the difference between fields is that, at least in astronomy, our journals are supported by the professional societies, like the American Astronomical Society, and not run explicitly for profit.
I should also add that in astronomy, the authors pay a significant amount of to the journal when a paper is published. It is usually in the range of about $100 per page.
Exactly. Arxiv is a revolutionary thing for math, physics, CS etc. Good luck getting life scientists to put their precious papers there. Most don't even pay the open access fee when the journal provides it.
Ecology has a mix of for-profit and non-profit journals, but those run by the professional societies aren't always much better. We only recently (within the past few months) won the right to post preprints on arXiv.
There are still many papers left behind paywalls. It does depend on the field you're in, as reflected by other comments.
I left academia years ago but continue to (try to) do research. Resources like scholar.google.com can often help find a free-to-read version of a paper, which is great. But access to a single key paper can make or break your progress. I've personally dropped many, many hours of work because of this problem - I couldn't reasonably access key literature that I needed.