The title is pretty much what I seek to know. Unless the other person is play-acting, one can often make out the eye of a living person has a shine-of-life to it; a cadaver does not. Similarly the gleam appears brighter when a person is feverish, or excited otherwise ... or has some infection in the eye itself.
What causes this gleam? Is it merely nervous electricity at work that disappears when the system is shutdown/crashed?
Not being an opthalmologist, if one happens along and gives a better/correct answer, listen to them.
Otherwise, let's take a moment to look at the eye:
What is not pictured here is the saline-like solution that keeps the conjuctiva moist. If we include the lubrication in all of the surfaces which interact with light, our list would look like this:
Out of that list, the Iris, Retina, and Fovea are designed to absorb light; not reflect it. The Vitreous humour merely serves as a medium for light to pass through and probably would not reflect light on its own. The same is true of both aqueous humours - they do not modify the light so much as serve the eye structurally.
That leaves us with the Lubrication, Conjuctiva, Cornea, and Lens - surfaces which also interact with light, but act to some extent to reflect or modify it as well. The Cornea is what's actually operated on when a person undergoes corrective surgery; a pattern is etched into the Cornea which redirects light similar to the person's previous prescription and giving them better vision.
Now, because you mention the difference between an excited person and presumably someone at their normal state, it's worth to note (when dealing specifically with the eye) a very common side-effect of psycho-active drugs and emotional states (particularly fear and interest/curiosity) is the change in pupil size when lighting conditions don't change:
The reflection is different depending on the size of the pupil, and I would venture to guess the pupil size difference between someone who's excited or manic (whose system is flooded with endorphins) and the person at normality is part of what you define as 'gleam.'
The other situation you draw a difference between is a dead person vs. a living person. Before rigor mortis sets in, the relaxed state of the ciliary body is to stress the suspensory ligaments - resulting in pupil dilation. In addition, and I think this is the most important distinction: dead people don't blink. The act of blinking spreads the lubrication on the surface of the conjuctiva, and can be mimicked with tear-drops for people with dry eyes. Because the lubrication acts as another surface to reflect light (similar to having a thin layer of water on a surface), the dead person's eyes will have less of a "gleam" than a living person's.
So, to sum up: The "gleam" in a person's eye is probably due to hydration and/or pupil dilation. Pupils will dilate at lower light levels, under the influence of certain drugs, or when showing interest. Hydration is constant, except when the lubrication is not produced in the quantities necessary or when the person is unable to blink ('dead' in your example).
