While doubtless some do actually sleep with the Good Book physically open by their bedsides, the bulk of the usage of this phrasing is idiomatic. At least in every context I've ever heard it, whether a Bible was physically open or not was not the point so much as that it was easily accessible and actively used: the last thing you were looking at each night and the first thing you would reach for in the morning.
Given that you say when you've heard it the references have been vague, I suspect what you've been hearing is also this idiomatic usage. To my knowledge none of the major branches of Christianity teach that this should literally happen. Again it's possible that some small groups here and there hold some eccentric view, but most English speaking Christians are less likely to mean this literally than idiomatically.