This would drive the first significant changes in OS architecture in decades. Up until now, OSes have largely relied on reboots to fix bitrot in system operations. What do you do when there is no more "rebooting," when system state persists across power cycles? The first solutions will probably be intentionally wiping blocks during first initializations, but I think this might actually drive more adoption of micro-kernel architectures, where the core is proved solid against state corruption, and everything else can be hot-reloaded when errors stack up enough to cause a core dump.
TL;DR: Right now, when Linux crashes, it's usually because some subsystem becomes corrupt, and Linux has very few ways of clearing these; think zombie processes, which can only be cleared by a reboot. With persistent memory, Linux will have to get better about this, or else a micro-kernel will have a chance to gain the upper hand.
Not for the first time do I wish someone with Linus' motivational and organizational skills would pick up MINIX and take it over the finish (Finnish?) line; it's so annoyingly close. But maybe Redox will get there.