Well, both WoW and D3 are loot based games -- at their hearts, you kill monsters for their loot in both. Diablo III's Loot 2.0 is an amazing system for itself -- it's fun and exciting when the loot you get from monsters ends up being awesome. But for WoW I think a system this personally rewarding would be insane. World of Warcraft is not set up for you to have some of the coolest, best items you'll ever see just drop from random trash, because WoW isn't a game designed around the idea of constantly going on reaping trash stomping dungeon crawls. The social aspect of WoW, with communal play of small 3, 5 and larger group content isn't the same as D3 and its multiplayer. If anything, Warlords' item squish is already a cautionary tale of the kind of excess that Diablo III not only rewards, but exults in. Nothing helps drive the lesson home that these are different games like their different approaches to loot.
That said, WoW can learn something from D3 in terms of the way loot works.
Diablo III has a lot more going on with its gear than WoW -- stats that don't have anything to do with direct, measurably character power, at least not in the way we're used to thinking about it. Stats that increase resource regeneration, buff individual class skills or powers, or even stranger (like the bracers I opened this post with) - we saw this embodied in the idea of tertiary stats at BlizzCon, and Diablo III shows us one paradigm of how to implement it. Even if WoW doesn't crib it wholesale (and I suspect they neither should nor will) it's useful to examine how the game does what it does.
Similarly, I don't think the concept of Paragon Levels is one that works well for WoW, but I do think Diablo III has a shared bank and gold system that WoW could benefit from. But for my tastes, the real gem of Diablo III is the random objectives. Simply discovering events in the world, taking part in them, and profiting in some way (gold, XP, loot) is brilliant and it's something I know we're going to see in Warlords, and frankly it's something we really need. I'd like to see the idea expanded - have events that spawn randomly in the world, in different locations, with changing elements (random spawns, different objectives each time) to keep them fresh.
I'd be lying if I didn't admit that I still prefer several of the D3 classes' resource generation/management systems to WoW - the Barbarian, Demon Hunter and Monk particularly. I'm aware that each is a separate game with separate needs, but the fact is, D3 feels more like your resource spenders are rewards, big flashy attacks that make you happy to use them. I'd like more of that feeling in WoW - getting and spending, we should lay waste our enemies.
The decline of the Real Money Auction House (indeed, the auction house as a whole is being removed from Diablo III as of March 18th) really highlights where WoW and Diablo III differ, and for each game's overall good, should remain different. That said, WoW could stand to have more gear that isn't soulbound - bind to account is one possibility (D3's legendary items, for instance, are account bound and I think that would be a nice feature for WoW's much more rare legendary items - my level 70 monk should be able to use my 90 warrior's Warglaives of Azzinoth)
I've already talked about how WoW should steal D3's version of transmog. That's not changed. Steal it. Steal it hard. In addition, one of the things I really like about Diablo III right now is how abilities and runes interact with gear -- there's specific gear that's good for one spell (my helmet increased my barbarian's Whirlwind damage, for instance) and as niche as that is, something that buffed all talents on a specific talent tier could be really nice for WoW. With the technology already in place so that there's no strength or holy plate, just plate, a piece of gear that just buffed the level 60 talents and changed description based on class could certainly be implemented.
In the end, the main lesson Reaper of Souls and the patch 2.0.1 update have to teach WoW is this - stay true to your core. The Loot 2.0 update works great for D3 because it fits in with the core philosophy and gameplay ethic of the series -- you go out, you kill stuff, you get loot for it, that loot is usually good for you. It fixes the main problem people had with Diablo III at launch - the gameplay felt strange with the emphasis on an element that wasn't core to the series like the Auction House. Going into Warlords, with all of its system changes and item squishing, the core of the World of Warcraft experience needs to be kept fully in mind at all times -- you don't need to capitulate to players and just give them everything they demand, but you do have to give them the game experience that you've trained them to expect. Patch 2.0.1 works (at least it did for me, running through with my wife last night) because that's exactly what it does -- it makes Diablo III a much more Diablo experience.
That's exactly what Warlords of Draenor needs to do. You got one patch very right, Blizzard -- now do it again.