I've tested this on a multiplayer server running 15w43a. Zombies would not pickup anything.
I tested once again on latest 15w46a on my own pc using Windows 7. The zombies will not equip armor or weapons on the floor. They will not hold any item I throw at them. I tested different modes survival and creative. To see if the tracking AI would trigger it. Still did not work. I tested naturally spawned zombies vs ones I placed with a spawn egg. None of them would pick up any items. I did notice that zombies are spawning with random armor as usual. They just won't pick anything up. ,