A forward C of G produces a nose down pitching moment. In order to maintain pitch attitude, this nose down moment must be balanced by a downward force from the tailplane (or upward force from the canards). Creating these forces produce trim drag, which decreases fuel efficiency and range.
As the C of G moves rearwards towards the neutral point, the nose down pitching moment decreases. This decreases the nose up trim force required, which in turn reduces trim drag and increases fuel efficieny and range.
this process continues until the C of G reaches the neutral point, at which stage the aircraft becomes neutrally stable. In this condition it generates no pitching moments and so requires no pitch trim. The fuel efficiency and range are (in theory) at their maximum at this point, but the handling is dreadful.
If the C of G continues to move aft, the aircraft becomes unstable and generates an increasing nose up pitching moment. This requires an increasing tail up trim force. The generation of this force produces trim drag, so fuel efficinecy and range decrease.
So fuel efficiency and range increase as the C of G moves towards the neutral point, and decrease as the C if G moves away from it. In conventional aircraft the entire C of G range is ahead of the neutral point, so aft C of G movement within authorised limits improves fuel efficiency and rage.
The short (increase or decrease) answer to your question depends on the relationship between the C of G and the neutral point.