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.