If those icons are placed there by an extension then you may not be able to move them to another toolbar via View > Toolbars > Customize.

Which extensions have placed items on the Add-ons Toolbar?

Yes they've placed by add-ons like Quick proxy and Ghostery. I even suggested Firefox team to add this ability to Firefox in beta period, but I think it would be so long to do it up! Is there any extension to access the Add-on bar?

I think I figured this out. Got to the toolbar that you want to pull these icons to and right click on an open space and select "customize". This will then open that box which contains icons that you can choose from to drag to that bar. Now look at the icons on your add-on bar and drag the ones that you want to that box vs directly to the toolbar. You can now drag those icons from the box to the toolbar. I did it with two extension icons that I wanted up top because having it on that stupid looking toolbar at the bottom looks lame. At least it does to me:)

@CraigMJamieson

That won't work for add-ons that use the "old Statusbar" coding for icons / buttons, which is what many of the buttons that are placed there automatically are "fashioned" by. Mozilla left some of the code necessary for that old way of doing buttons, so as not to break older extensions that weren't updated completely or correctly for the new Add-on Bar with true Toolbar buttons. Eventually that old code will be removed, it's just a way of "grandfathering" for the old way it was done for so long.