This line is provided by the wrap-guide package. You have the following options to remove the line:

Option 1

Disable the wrap-guide package. Go to Atom > Preferences > Packages, type “wrap” into the search box, then wait for the wrap-guide package to show up under Core Packages, and click the Disable button there.

Option 2

Or add the following to your user stylesheet. Use Atom > Open Your Stylesheet to open the styles.less file in the editor, then add the following:

atom-text-editor::shadow {
    .wrap-guide {
        visibility: hidden;
    }
}

Once you save the file, the line will be gone.

More info can be found here: https://discuss.atom.io/t/vertical-line-in-editor/2717

Update 2015-06-28 Updated selector due to updates in the Atom style classes. The above now works in Atom 1.0+.