CG vs. fuel consumption
 

I know that the farther aft (closer to Centre of Pressure) CG moves, the less fuel is burnt because of less downforce req'd from the tailplane.
But is it (in theory) better to have the CG at CP, rather than aft of the CP? Would the trim- and induced drag of the tailplane outweigh the benefits of producing positive lift if CG is located beind CP?

Thanks!

It depends on how the wing span load is distributed, but in general you want as little load as possible carried by the trimming surface.
This applies to both horizontal stabs and canards because:

1) The lifting surface with the most span is best at minimizing drag due to lift(induced drag).

2) A smooth total span load distribution is best for low induced drag. You don't want trimming surface lumps.

CG aft or at the CP? Wouldn't that make for a REAL unstable a/c? ;)

CG aft of the CP, or the AC (aerodynamic centre), isn't a stability issue.

Longitudinal stability is determined by the static margin (cg to neutral point) and manoeuvre margin (cg to maneouvre point). Both of these are determined for the entire aircraft configuration i.e. "tail on".

Normally one talks of the CP or AC as "tail off".

The NP/MP are "always" aft of the AC, so the CG may be aft of the AC/CP and the aircraft remains stable.

25% MAC is a good approximation for the CP of the wing (that's why it came into use), and the fuselage tends to pull the wing-body CP/AC forwards. Yet many aircraft have an aft cg limit well past 25% MAC, with no stability issues, because the tail is doing what it's there for.