An invisible man would have transparent retinas — and thus be blind.

UPDATE: Wells seems to have thought of this! In Chapter XX, shortly after his transformation, the Invisible Man says:

“I struggled up. At first I was as incapable as a swathed infant — stepping with limbs I could not see. I was weak and very hungry. I went and stared at nothing in my shaving-glass, at nothing save where an attenuated pigment still remained behind the retina of my eyes, fainter than mist. I had to hang on to the table and press my forehead against the glass.”

In testing his cat, he had found that “there remained two little ghosts of her eyes … the back part of the eye, tough, iridescent stuff it is, wouldn’t go at all.” (Thanks, Nathaniel.)