Checked with my wife who is an optometrist, and she said cornea.

Edit:

She added, "cornea has more nerve endings"

Edit 2:

She also added that the inside of your eyelid is a mucus membrane, so soap is no more irritating to it than it would be to the inside of your mouth.

It's a similar burning feeling to keeping your eyes open for too long; so, the eye balls I would think.

The cornea should be one of the most sensitive areas (in terms of irritation) of your eye.
I don't know, but if I had to guess, I'd say it plays a crucial role in feeling that burn.

There are already some excellent scientific answers here.

It's quite easy to verify where the burning is. Notice a spot with more burning and try moving your eyelids (open, close) and then moving your eyeball (left, right, up etc).

The area that burns moves with the eyeball as if it's on the surface.

the skin around the eyes is very thin, soap is very harsh on thin skin areas.