There's nothing as difficult as a close-up.In his book Acting in Film, Michael Caine describes a technique for the use of the eyes on camera and, specifically, on close-ups.
I'll try to re-explain it in my own words since in the book I didn't find it very clear. I only fully understood what Michael Caine meant when I watched a documentary about a series of classes he held to a group of extremely priviledged acting students.
Michael Caine suggests that, when on a close-up and addressing your lines at an off-camera actor, the leading eye should be the one closer to the off-camera actor. In other words, if the off-camera actor stands to the right of the camera, the on-camera actor's leading eye will be the left one; if the off-camera actor stands to the left of the camera, the on-camera actor's leading eye will be the right one.
Now, why such a fuss?
First, choosing a leading eye prevents you from shifting involuntarily from one eye to another. Such movement wouldn't be noticed in real life, but a close-up will emphasize it to a point where the micromovement of the eye will then become evident and distracting.
Secondly, choosing as a leading eye the one closer to the off-camera actor will put your face in a very advantageous position with respect to the camera: as much as possible of your face will be shown.
Now, a word on the off-camera actor.
In my experience, especially when taking on-film acting classes, the off-camera actors never give as much energy as they gave during their own close-up. They often read the lines even though they're off-book; they only half react - when they react at all - to the on-camera actor's performance, often never keeping eye-contact.
Personally, I find this very selfish and disturbing.
A movie is a collective project. As it is true that an actor should give the best of himself and just as it is true that he should do so for the sake of his own performance, he should first and foremost do so for the sake of the whole movie. This means that actors are not hired to score; they're hired to cooperate with their fellow actors in the movie for it to become a product of high artistic quality.
Lastly, if you're the off-camera actor, stand (or sit) as close as possible to the camera. I don't want to have to give the camera my profile to speak to you and, thus, have my close-up ruined.