the magic eye is a photo electric cell which is used for flame supervison, i am not sure about this burner but most if not all have them, as the detect flame failure, if yours is dirty, it may not be "seeing" the flame and thus locks out, alos if they move they can cause problems,
I wouldnt be so sure that cleaning it will make your burner fire, many things such as control box, oil pressure etc all can cause an issue
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