Everybody seems to agree that students are really motivated by grades, and if reading the material was high stakes enough, they would have an incentive to do it.
I'm not saying this wouldn't be a bit of work. In order to prevent students from just exchanging text files as to what the answers were, you'd have to use the Random Section feature of the D2L quiz tool (I'm sure other LMS's have this feature) to deliver ten questions out of a bank of about thirty to each student. From what I've seen of publishers' test banks, this isn't really unreasonable if the reading was a chapter, but if the reading wasn't in a publisher test bank, somebody would have to write it. Since a lot of the classes where this is a problem (cough, general ed, cough) are probably taught by several instructors, you could probably divide up the load and come up with a significant bank of questions without any one person having to do all the work (also sounds like great duty for a grad assistant).
One objection that I could think of is students could get together and work out the answers together, but that might not be such a bad thing. If a few laggards in a group depended on the smart kids to just find the answers, that would probably be pretty obvious to the group and the laggards would be, let's say, subjected to group pressure.
Maybe this has been tried. I'd be curious to hear about it.