Deleting the rim shadow node will not remove the rim shadow - it will maximize it. That isn't the node that does the rim shadow. That is the node plugged into the node(s) that make the shadow. By deleting it, you're setting it to 1, or maximum because somewhere else the maximum value is multipied with this value, by virtue of being connected to it. Connecting a node to an input multiplies that node with that input. When you delete this node, the input is left with its full value, instead of being modulated by the PM:Rim Shadow node.

Just so we're clear, I'll say again that the rim shadow effect is not a real shadow. It's node math. You also said you made sure cast shadows is off for the cover, which is good. Otherwise it really will cast a shadow, but not just a rim shadow - a full shadow of the entire cover.

As an experiment, move the eye cover forward, away from the iris, by about .1 inch and see what it looks like. If the dark ring goes away, this is a ray bias problem.

You could also try just rendering the eye cover in front of a white card.

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)