Sympathetic nerves control the tiny muscles that cause the pupil to dilate in darkness or during excitement and control the muscles that hold the eye forward in the eye socket. That's why when sympathetic innervation is blocked, the eye will sink into the orbit. When this occurs, the eyeball can't keep the third eyelid retracted as it usually does, so it becomes elevated. The upper eyelid will droop too. Finally, without sympathetic nerves, the pupil can't dilate, so you'll see a very small pupil, even when it is dark.
Damage can occur anywhere along the nerve as it travels from the brain to the eye.
The good news is that Horner's syndrome doesn't hurt. The eye condition is not painful. It's mainly a cosmetic problem that typically clears up in time on its own. For more information about your pet's eyes, contact your local veterinarian.