That would be a task of compositing using something like Nuke or After-Effects. Here's how I'd tackle something like this:
- Grab a frame of the person looking directly in the camera
- Track the positions of the eyes ("track_left" and "track_right" as seperates)
- Mask out the eyes from the still-frame and separate them into eye_left and eye_right
- Link each eye to the corresponding tracking, so that it follows the original eyes.
- Perhaps apply a subtle slow flickering (add curves and in the effect-opacity type in the expression "wiggle(0.5, 15)" or something in that area, using After-Effects)
- When the person is looking down, try to keyframe the opacity of the eyes so the real eyes are revealed from time to time, depending on how convincing this turns out.
If the tracker doesn't stick very well you can try the mocha-tracker which comes natively with After-Effects for the latest versions. Make sure to only mask out the inner area of the eyes - don't patch out too much of the eyes and only the bare minimum to remain realistic shadows, wrinkles, etc.
I hope this works out for you - if not, please post your results and we'll see how to improve it.