Lets take a look at a few potential criticisms that I have of the paper.
1. The paper argues that
"First, numerous studies show that stock market prices do not follow a random walk and can indeed to some degree be predicted ... thereby calling into question EMH’s basic assumptions"
Out of the papers that are cited to support this, only one (#8) is actually published in a journal that I have heard of and that paper is referring to thinly trading stocks. Journal quality matters and it matters a whole lot when we are looking at such a fundamental theory as the EMH. If the authors had cited numerous studies from leading finance journals, then they would have been on solid ground. The fact that they didn't speaks very loudly. This is either shoddy authorship, or the work of someone who just doesn't understand the subject. The evidence for the EMH is very, very strong.
2. The paper claims that there is evidence of online chatter predicting other things.
"Recent research suggests that news may be unpredictable but that very early indicators can be extracted from online social media (blogs, Twitter feeds, etc) to predict changes in various economic and commercial indicators. This may conceivably also be the case for the stock market. For example,  shows how online chat activity predicts book sales.  uses assessments of blog sentiment to predict movie sales."
Again, the authors demonstrate their complete lack of understanding of the EMH hypothesis. There is no surprise that online buzz is related to movie sales. The EMH states that you cannot use publicly available information to earn a profit above the risk adjusted rate of return. It doesn't say you can't predict how well a movie would do.
3. There is a long list of things that are supposed to predict the market, that include: super bowl winners, hemlines and sunspots. In all cases, these are confusing correlation with causation.
4. The paper doesn't do an out of sample test. Any trading rule that is found to be revealed in past data must be then tested in new data.
5. The paper ignores trading costs. What are the potential returns that could be earned by trading on this information? What are the costs of implementing the strategy?
6. The results could be driven by a few events. For example, because the markets are not open on weekends and if bad news originated on the weekend and then was twittered (tweated?) over the weekend, it would appear that you could predict returns. However, because the markets are closed, there is no way that the bad news could be impounded in to prices. Thus the illusion of predictably would be created. There is evidence that in 2008 a lot of bad news happened on the weekends.
7. Finally, let's just assume for a minute that these authors had found a way to reliably predict the Dow 30 using publicly available information, and could do so and make a profit. By publishing this information, they basically rule out any chance of making massive profits from their research.
Unfortunately, this paper is on the verge of going viral. A google search of "can twitter predict the stock market" yields 348,000 hits. However, a search of "can justin bieber predict the stock market" yields 968,000 hits. No doubt, it will be picked up and written about in numerous news outlets (see my earlier post on how well journalists understand finance).