Based upon the updates to the question:
If your characters need to orient themselves in a particular direction before they start walking, it will be very difficult for them. You've removed all the things that might typically be used. None of the following are available:
- Stars at night
- Magnetic fields
- Natural landmarks
- Artificial landmarks
- Water features
- Predictable/persistent weather patterns
Thus I'm not sure of any practical orientation technique that would allow them to set out on a specific heading. Though perhaps that's not required?
If it is, maybe at night they can see a faint glow on the horizon in the general direction of the nearest settlement. Or maybe there are auroras or some other aspect of the fantasy world that would be usable to get a ballpark directional fix. Otherwise...I don't see how your characters can orient on a specific heading with no visual references whatsoever, unless they have a sixth sense or a very keen sense of smell or similar contrivance.
Maintaining a Heading
This is easier based upon the scenario you've described. The trees in your forest aren't that large or that dense, they're just homogeneous and cover a large, flat area with no built-in visual references.
In that situation, all your characters need to keep a straight heading is a bit of wire, a knife or other sharp tool for making marks (optional), and some math. Your character should make a loop out of the wire, and attach it to their headgear so that it sits in front of one eye. They can then line their makeshift targeting reticle up with a distant tree and walk towards it. They'll travel in a straight line as long as they keep the tree centered in the reticle. And they should keep a count of the total number of steps taken to reach the tree.
When your character reaches the tree, they should mark their inbound trajectory with the knife (for instance, in the ground leading up to the tree, or the tree itself). Continuing on that trajectory they should proceed to the far side of the tree and look for a new tree in the distance to use as the new reference point. Before they go anywhere, they should mark the angle to the new reference point, and take note of the angle by which it deviates from their inbound course (I'm assuming that the trees won't always allow for a perfectly straight path to be followed).
Now all they have to do is proceed to the next tree, and repeat the process. By tracking the angle of each course deviation and the distance traveled between each tree, your scientist character should have no problems holding to a straight course by making minor corrections when choosing the next reference point (i.e. if you deviated to the right by 5 degrees for 500 steps, you'll want to compensate left 5 degrees relative to the original heading for 500 steps).
That should allow them to maintain a reasonably straight heading over a relatively long distance. Or if you give your characters very accurate tools for measuring distance and angles (a pedometer and a protractor, maybe?), it should keep a straight heading essentially indefinitely as long as there are trees to use as reference points. They could even execute turns and other maneuvers to detour around problematic bits of terrain if need be, just by keeping track of the angle and distance of travel. It's essentially a poor-man's inertial navigation system.
But won't work if, for instance, they get chased by something and lose track of either bit of information.