It's interesting, I think many people's first reaction to this would be "not in our lifetime" but this question actually reminds me of a story about my niece doing her homework one night. She was doing one of those exercises where you have to find a list of objects in a busy picture of a house and circle them. She got them all done except one, she tried and tried but couldn't find it. It's not that it was hidden particularly well it was because the object she was looking for was a tape recorder and believe it or not most kids today have no idea what a tape recorder is.
As I discussed in another post, the tools and resources are available to go paperless. In my opinion the roadblocks to going paperless are costs, integration and a willingness to change. All of these roadblocks are quickly beginning to change. The cost of these paperless systems are dropping as technology advances, there is a whole new industry evolving around integrating key business systems and the Gen Y's and Millennials have grown up with technology in their hands - using paper is a change for them.
While it's difficult to predict if paperwork will take a similar timeline as the "extinction" of the tape recorder, my best estimation would be one to two generations past Millennials. They will be the ones who heavily implement the paperless office and shortly there after people will be baffled that we used to use pen and paper.