Under Reg E are banks required to keep a Error Resolution log?

No, but I don't know how else you would track and test compliance? Most banks keep a pretty basic complaint log in excel with generic columns with titles such as "complaint date," "Customer Name/Account Number," "Card #," "Amount," "Type of Transaction," "Received Date," "Provisional Credit Date," "Resolution Date," etc.

Let me add that keeping a log is also a great way to easily quantify losses, determine investigation resources, and create budgets.

Thanks