Are open access papers read by a larger readership than paywalled papers?
This answers is not exactly what you ask for — you ask for readership, but most of the research has focused on citations. The two are, of course, related, and the answer seems to be a pretty clear yes. (In my field, it's practically the only way I find stuff from overseas, as many online repositories don't seem to have much international coverage except for a handful of big general journals but not the niche ones).


Gargouri, Yassine. “Self-Selected or Mandated, Open Access Increases Citation Impact for Higher Quality Research.” Public Library of Science ONE 5.10 (2010): n.p. Web.
This article indicates that citations are more common for open access articles, and that this is a result of higher availability and access (=likely readership) by researchers. [Link]
Lawrence, Steve. “Free online availability substantially increases a paper’s impact.” Nature 411 (2001): 521. Print.
This comes to the same conclusion — way back in 2001. The effect is found to be quite massive: “Averaging the percentage increase across 1,494 venues containing at least five offline and five online articles results in an average of 336% (median 158%) more citations to online computer-science articles compared with offline articles published in the same venue.” [Link]