Sorry, the above reply is wrong. Your visitors don't see the "Edit" tab: only you see it when logged in. "Leave A Reply" goes away when you disable commenting.
Disabling commenting in Settings > Discussion applies to posts and pages created after you make that change. For already published posts and pages you need to edit them and disable comments and pings in the Discussion module of the editor. If you're not seeing the Discussion module while on the post/page editor screen, click Screen Options (top right) to activate it. You can also bulk edit them: you go to Posts > All Posts, or Pages > All Pages, tick the square at the left of "Title" to select all, select Edit from the Bulk Actions pulldown, click Apply, select "Do not allow" from the Comments pulldown, same thing from the Pings pulldown (posts only), click Update.