Are there any fly-by-wire airliners with negative or near-neutral pitch stability? Are any modern commercial airliners with fly-by-wire flight control systems designed with negative or near-neutral pitch stability so that they can take advantage of the capability of such systems to reduce trim drag and increase fuel efficiency?

If so, which aircraft are so designed?

And if not, why not?



In the images, the aircraft with positive static stability must produce sufficient wing Lift (the green arrow), not only to hold the aircraft up, but an additional amount to compensate for the lift from the tail surface, which is pulling the aircraft towards the dirt. The total aerodynamic force is the sum of the Wing Lift plus the tail plane lift, which must equal the weight of the aircraft plus twice the lift produced by the tail. THIS MEANS THAT IN AN AIRCRAFT WITH POSITIVE STATIC STABILITY, THERE IS SIGNIFICANTLY MORE TRIM DRAG AND THRUST REQUIRED.

In an aircraft designed with negative static stability, (bottom image) The wings, (Green arrow) plus the tail plane, (blue) both contribute to positive (upwards) lift, and so the total aerodynamic force is less, therefore the induced drag is less, and the thrust required (and fuel flow to generate that thrust) is reduced in comparison to an aircraft with positive static stability..
if you would like more details, review this video, from which I quote (at about the 1:00 minute mark), speaking about general aviation aircraft with positive static stability,  "... Something not very pretty about this situation, [positive static stability], is that we have this downwards acting lift force[ at the tail], which we call Trim Drag, which essentially, in a negative way, impacts our range and endurance, and which we'd like to avoid...", and then later, talking about the negatively stable F-16, at 1:50 minute mark, "In this situation we have, [at the tail], an upwards facing Lift force. This means ... we have, as it should be, all the lift facing UP. This means [my italics], we have an increase in our range and endurance performance.." 

And, in addition, from this link, I quote "unstable designs get a slight reduction in drag and a slight increase in lift".  The rest of this article is also very informative.

To further illustrate the difference, below are two photos, of aircraft on final approach to land, where the flight path is stable and descending.
1. An F-4, with positive static stability, and 2. an F-16, which has negative static stability. Note the relative positions of the stabilators on the two aircraft, and especially, the relative angle of attack of the stabilators as compared with the AOA of the wings. In both of these pictures the aircraft is descending to land, so the relative wind is from the left, and below.
No.

Modern FBW airliners use less static stability than what the early jets were used to, but stability is still positive. The negative camber at the root airfoil of sweptback horizontal tails might indicate predominantly negative tail loads, but is also used to keep the isobars on the swept surface parallel. Also, the bent-up leading edge delays separation with full wing flaps when the tail has to produce lots of downforce:



A380 tail (picture source)



Boeing 787 tail (picture source)

At zero static stability the lengthwise position of the center of gravity coincides with the neutral point such that the lift per area on all horizontal surfaces is roughly the same (I neglect camber effects for the moment which are small in cruise configuration).

The horizontal tails even of the most recent designs still have negative camber and no twist which indicates that their design lift coefficient is around zero or even slightly negative. This has mostly to do with the tail loads in landing configuration but also shows that at transsonic speed they fly at a lower lift coefficient than the wing. Such an airfoil camber would make operation at transsonic speed and the lift coefficient of the wing rather inefficient.

A neutral stability design would make the FBW system even more flight critical than it is already and would make the "direct law" mode of the Airbus FBW system rather dangerous. Certification would be much harder - after all, FAR $25.173 still applies:


  (a) A pull must be required to obtain and maintain speeds below the specified trim speed, and a push must be required to obtain and maintain speeds above the specified trim speed.


The "maintain" part of this rule can best be achieved with positive static stability.

Also, reducing static stability beyond a point where stability is still slightly positive does not improve efficiency anymore. Note that the tail surface aspect ratio is smaller than that of the wing. If both surfaces would fly at the same lift coefficient, the induced drag from the tail would be relatively higher (in relation to lift) than that of the wing, even without downwash effects. This means that the lift at the tail comes at a higher price and it is more efficient to fly the wing at a higher lift coefficient than the tail.

EDIT:

Supersonic aircraft benefit from negative static stability because this will cause less trim drag in supersonic flight. No such advantage exists for airliners because they are not made for sustained supersonic operation.

Also, positive static stability is possible with lift on the tail. In that respect the question repeats an old meme. It is not the sign of the lift on the tail surface which determines the boundary between positive and negative stability, but the relative lift per area. It must be less on the tail than on the wing for stability. How else would a stable canard airplane be able to produce lift on its main wing?