Are there any fly-by-wire airliners with negative or near-neutral pitch stability?
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?