An "Invisible Man" in the Ralph Ellison sense-a social non-entity-can see, as we learn in his novel, all too clearly. An Invisible Man in the H.G. Wells sense of somebody who is "see-through" cannot, for two reasons.
The first and most quotidian reason is that, in order to form an image of his surroundings, the man's eyes must contain lenses or pinholes. Pinholes are obviously visible; lenses are visible because they distort the path of light passing through them, like the density variations in air rising off of the road on a hot day.
There is a way around this, however. Suppose that the Invisible Man wore a suit, crafted of unobtanium, that acted as billions of microscopic cameras and projectors, so that each little patch recorded what it saw and nearly instantaneously transmitted that to patches on the other side which would project the incoming light. Fully enclosed in this (hot!) suit, the Invisible Man could also have an image projected to his eyes, with his eyes remaining nonetheless hidden.
Fundamental quantum physics of light gets in the way: a Heisenberg-type uncertainty principle ensures that he will be seen. In everyday life, unless we look at a hologram or use a CD player, we think of light as being "incoherent" and thus described by its amplitude (brightness) and wavelength alone. But light also has wavelike properties, and like all waves, has a phase, which says what part of the wave cycle (crest, trough, or in between) a particular bit of the field is in. By shining a laser beam through two slits placed close together, one sees immediate evidence of this wavelike property.
Just as one cannot measure the position and momentum of a massive particle simultaneously and with infinite precision, one cannot measure the number of photons (amplitude) and phase precisely with the same device. This is called the number-phase uncertainty relation. The Invisible Man's suit, even if made of unobtanium, cannot in our universe ever know both how brightly and with what phase to project light out of its other side. If it gets the brightness right, the phase will be all wrong. If Mr. Invisible is being illuminated by an incoherent source, such as a light-bulb, this is of no concern, but if he crosses a laser beam with a known diffraction pattern-let's say he comes between a grating and a screen on which one is seeing a pattern-he's as visible as you and me!
H.G. Wells couldn't have known this, of course: the quantum theory of light wasn't worked out until the 1950s and 1960s!