Neither of them would sound wrong to me, but the second sentence may sound a bit more natural in this context, because the person is performing an action. In other words, you're basically asking "let me know when YOU have finished completing it". "Complete", to me, would sound more natural when just describing a noun in general. For example: "The painting is not complete." In other words, part of the painting is missing, or it is not a whole painting. "The painting is not completed" might imply that someone is in the process of completing it, but they are not yet finished. These are very subtle differences, however, and they can practically be used interchangeably. Quite a long answer, but hope it offers some insight :)

Both are perfectly fine, use which ever you prefer!

Just say, "Please let me know when it's done."