Do horizontal stabiliser force direction change with CG variation for a given aircraft?
The ideal cruise condition, which minimises induced drag, is for the tail to exert zero lift. This avoids any wasteful tip vortices from the tail.

However for most planes the CG shifts during flight, if only because fuel is being used up. Also, trim conditions change with speed, rate of climb, altitude and a host of other variables during flight. The tail spends most of its time a little away from the ideal zero-lift condition, and it can go either way.

The "lifting tail" was much discussed in the early years of aviation. It has a destabilising effect which limits the stable CG range and every plane has a maximum safe level of lift from the tail while remaining stable. Nevertheless, away from that limit it contributes lift which helps to offload the wing and allows it to be that bit smaller. 
According to Berriman; Aviation, Methuen, 1913.

"On some machines the tail planes are cambered like the wings and help to support the weight in flight. The angle of incidence of the tail is less than that of the wings, however, and the principle [of weathercock stability] apples equally to such cases." -- Page 69.

Berriman gives some supporting analysis in terms of CG and CP to explain what he summarises as weathercock stability. The subtleties of the lifting tail would be further elaborated in subsequent years.

The bigger the tail, the greater the CG range can be accommodated, and also the greater the temptation to offload the wing a little. The limit is the tandem-wing aircraft, where for stability the fore wing is loaded more heavily than the aft, but the aft still contributes substantially to the lift under all flight conditions. A tandem-wing modification of the Westland Lysander flew experimentally during WWII and test pilot Harald Penrose confirmed that its handling was more stable than the conventional version and had a greater CG range.

So ultimately, every plane has its own practical tail loading, which may be normally up, normally down, or variable around the zero point.

There is a myth that for a plane to be stable the tail must always exert a downforce. This seems to arise among students for two reasons. One is that, when the AoA increases, dynamically it must exert a larger relative change in downward moment about the aerodynamic center than the wing's upward moment. Diagrams showing this effect always have a large down-arrow at the tail and it is easy for the student to forget that this is a relative change not an absolute value. The other is that the wing works hardest at takeoff, when fully laden and at its slowest flying speed; when the tail pushes sharply down to rotate the nose up, the wing has to compensate for that as well, so this is the design condition for sizing the wing and everybody has to learn it. Put that first diagram alongside and you have a myth in the making. If you failed to understand any of this paragraph, then you too may fall for it!