For those not familiar, the mass market paperback is that chunky, newsprinty $4 to $8 paperback you find in airports and grocery stores (in addition to traditional independent and chain bookstores). It's especially popular with genre fiction (fantasy, mystery, romance, science fiction). Historically, they've sold well because they're cheap, lightweight, and don't take up a lot of space; not many people buy them to display on bookshelves or coffee tables.
E-books are already relatively cheap, and they have no weight and occupy no physical space at all. As the cost and heft of e-readers steadily declines, there will be (in my opinion) no reason to buy a mass market paperback rather than an e-book, and I think this will lead to the format's demise.
A lot of people are currently worried that e-books will kill the hardcover, but I find this relatively unlikely. Hardcovers have been status symbols and conversation pieces for centuries, if not millennia. People like having bookshelves full of hardcovers. They like having them signed. They like physically perusing a library rather than flipping through a list of titles on a screen. For these reasons (among others), I think hardcovers will survive the conversion to e-books, although I certainly expect print runs to be reduced and POD to become a more tenable option for smaller publishers.
As for the mass market paperback: granted, the lendability factor will definitely keep it alive for a few more years, and it will probably take decades beyond that before the second-hand market begins to fold. As soon as solid lending or renting protocols are established by the e-book industry, however, I don't see any reason why consumers would rather have a physical, low-quality paperback than a non-physical, high-quality e-book. Can you?