Since we're alreaedy sharing the showComments in the if section, how do you feel about removing some duplication? 
if (showComments) {
  let moveHeight = height;
  let moveWidth = width;
  if (inMobileMode) {
    moveHeight = CommentDialogView.COMMENTS_BOX_HEIGHT_MOBILE;
    draftFormY = ...;
    ...
  } else {
    moveWidth = CommentDialogView.COMMENTS_BOX_WIDTH;
    commentsWidth = ...;
    ...
  }
  this._$commentsPane
                .outerWidth(moveWidth)
                .outerHeight(moveHeight)
                .move(0, 0, 'absolute');

  const $commentsList = this.commentsList.$el;
            $commentsList.height(this._$commentsPane.height() -
                                 $commentsList.position().top);
}

Not too sure if the order of the commentsPane matters though

Can you check what happens on a mobile device in landscape mode?

We might as well optimistically change this to use the new var(--ink-z-menu)

And var(--ink-z-dialog) here.

Instead of "mobile" vs "not mobile", perhaps we should call this portrait vs landscape layout?

We might as well just call this inline, since we only use it once.

Missing Returns:

This needs a type annotation.

Is there any reason not to make this #private?

Since we're already defining height here, how do you feel about just assigning it to CommentDialogView.DIALOG_NON_EDITABLE_HEIGHT?

Ship It!

I noticed in the screenshots that the margin around the other-comments box is too small on the bottom. Can we add box-sizing: border-box to this, to fix that?

There's an extra space before the :