So, I was thinking about why the eyes of those who are killed by shardblades burn out.
Eyes are often described as the doors or the windows to the soul and if I remember correctly, shardblades sever the connection from the physical to the cognitive realm (where the "soul" is located). When killed by a shardblade, it severs your connection to your soul and thus the eyes burn out because that window to your soul doesn't lead to your soul anymore.
I don't know if this makes any sense, but I found it interesting to think about.