Q:

How do I turn off the border around Unity launcher icons?

Is there a way to turn off the boxes around the icons in Unity?
Clarification: I do not mean the backlight color -- there are plenty of questions about turning that off. I want the icons, and just the icons, to appear in the Unity bar, without any lights and shadows and borders. Can this be done?

A:

The launcher icon overlays such as borders and highlights can be removed quite easily by changing 4 icon overlays in Unity.

Make a backup of the following image files in the folder
/usr/share/unity/4/:
launcher_icon_back_54.png
launcher_icon_edge_54.png
launcher_icon_glow_62.png
launcher_icon_shine_54.png

Either make your own transparent images of the same size or download the files at
wget -O transparent54.png "https://i.stack.imgur.com/MiFpK.png"
wget -O transparent62.png "https://i.stack.imgur.com/0NvP0.png"

Replace those 4 files with the same 4 files downloaded from the above link.
Restart unity by pressing Alt+F2 and typing setsid unity (or logging out and in again).
If you don't like it then just replace the icons with the backed up icons and restart unity. 

Example:

A:

It is possible, and rather simple.

Install compizconfig-settings-manager:
sudo apt-get install compizconfig-settings-manager

Then run about:config in the run dialogue in Unity: press Alt+F2 and enter about:config.
CCSM will pop up with the Unity plug-in open. Navigate to the experimental tab:

Now change the Backlight Mode to "Backlight Always Off".
Problem solved.

In light of your comment, the box never 100% goes away (by design).
Small note from @ccpizza concerning 16.04.1.

In ubuntu xenial 16.04.1 the Experimental tab is gone, and the setting
  is now under the Launcher tab and the dropdown is called Launcher Icon
  Backlight Mode.

A:

I scripted the removal of the boxes around the applications and the one around the Ubuntu button (for 12.04):
cd /usr/share/unity/5
wget -O transparent54.png "https://i.stack.imgur.com/MiFpK.png"
wget -O transparent62.png "https://i.stack.imgur.com/0NvP0.png"

for name in squircle_shine squircle_edge squircle_base squircle_base_selected launcher_icon_back launcher_icon_edge launcher_icon_shine
do 
    sudo mv ${name}_54.png ${name}_54.ori.png \
        && sudo ln -s transparent54.png ${name}_54.png
done

for name in squircle_shadow launcher_icon_glow launcher_icon_shadow
do 
    sudo mv ${name}_62.png ${name}_62.ori.png \
        && sudo ln -s transparent62.png ${name}_62.png
done

Then to restart Unity, press Alt+F2 and run setsid unity. The result looks much cleaner to me. 
Before: 

After:

To switch back:
cd /usr/share/unity/5
for name in squircle_shine squircle_edge squircle_base squircle_base_selected launcher_icon_back launcher_icon_edge launcher_icon_shine
do 
    sudo rm ${name}_54.png \
        && sudo mv ${name}_54.ori.png ${name}_54.png
done

for name in squircle_shadow launcher_icon_glow launcher_icon_shadow
do 
    sudo rm ${name}_62.png \
        && sudo mv ${name}_62.ori.png ${name}_62.png
done

