Do ducted fans eliminate induced drag?
EDIT: I'll add this ahead of the answer because it will necessarily come up: This answer is based on the model described by Prandtl, which is only that: A model, a simplified description for what happens that works to explain some phenomena. There is some disagreement about this model, so a separate explanation using a differen theory might help. The problem is that an explanation with pressure differences doesn't help much here, at least I wouldn't know how to answer the question without working with Helmholtz/Prandtl.

Fundamentally, what we are talking about are vortex rings that are created because air is moved in some direction somewhere and not moved everywhere else. That's independent of how you move it, and fundamentally, a duct or winglet or something similar can't change that. You can look at the vortex itself as the thing that actually creates lift or thrust, so you want and need that vortex. 

Then there's Helmholtz's theorems. Quick quote from that article:


Helmholtz’s first theorem:
The strength of a vortex filament is constant along its length.
Helmholtz’s second theorem:
A vortex filament cannot end in a fluid; it must extend to the
boundaries of the fluid or form a closed path.


In the case of a normal wing, the vortex ring ("closed path") consists of one side along the length of the wing, two sides trailing behind the wingtips and the fourth side is somewhat lost, or far enought away that it can mostly be ignored. The three sides that are not along the wing itself are the problem, they are what the term "induced drag" refers to. Because there is no surface at the end of the wing for the vortex to end at, it has to continue into infinity. And there is some friction involved in creating long fast-spinning vortices, so you get drag. (Any help in explaining this part more eloquently is appreciated)

Because the amount of lift/thrust that you want is given, the vortex strength isn't something that can be changed either. What can be changed, however, is the distribution/speed/radius. If the lift distribution is elliptical (see: Spitfire), the lift gradient is minimal across the wing so there is no strong and fast vortex at the end, it is gradually created along the whole wingspan and the induced drag is minimal. Winglets do something similar by forcing the vortex to go all the way around the winglet.

In the case of a propeller, the inner side of the vortex ends at the spinner or shaft, so the ring is broken and you are only left with a linear vortex, but it is still trailing infinitely into the air at the tips of the blades. You can look at a wing in the same way if you interrupt the ring at the fuselage.

In the case of a ducted fan, you have two fluid boundaries, both on the inside and on the outside, which appears to solve the problem. You can have a vortex around the blade only and no vortices trailing into infinity creating drag. The problem is that the fluid doesn't really end at the duct because the duct has a finite length. So what you get is a large vortex around the whole duct. Air is moving in at the front and out at the back, and somewhere, that air moves back to the front far away from the engine. 

In the case of a ducted fan, the vortex ring is created in the place where the moving air meets the static air at the beginning and end of the duct. Instead of a very small vortex around the tip of the wing (and along the wing, to not mess up Helmholtz's theorems), you get a larger one around the whole duct. 

Therefore, I wouldn't say that the duct can eliminate induced drag. The effect is fundamentally comparable to a wingtip device: It increases the inner radius of the vortex and thereby reduces the induced drag significantly. It's far more effective at that than a wingtip device, but it has to be because ducted fans are usually relatively small so the drag would otherwise be enormous.