I use konsole with tabs for multiple screens. Each window requires up to 20 tabs with unique names. The icon wastes valuable real estate. I want to remove the icon.

I know where the profiles are saved:

/root/.kde/share/apps/konsole/profiles

And in the profile file there is an entry for icons. I have tried different variations to remove it but no luck:

Icon0=none
Icon1=""
Icon2=null
Icon3=openterm
Icon4=openterm
Icon5=openterm
Icon6=openterm
Icon7=accessibility-keyboard-capplet

Is there a setup variable that controls this?

Also what is the default location for the icons? For worst case I can create unique icons with text to identify each tab. Very tedious for all the tabs I need.

Short answer: Patch the source.

Displaying icons in all kind of DEs follows usually somer rules from freedesktop.org.

In order not to have to code each icon again and again, freedesktop defines a set of icon-names for all that widgets, actions and so on.
The purpose of this is, to get a consistent user experience and make life easier for programmers.

If the programmer chooses to use that naming-scheme, there is no more call to display a specific icon in the source anymore.
The icon will be loaded from the DesktopEnvironment from the default icon scheme (which names the file or files of that icon).

Apperently Konsole does not offer a choice not to display this icon.
So the only way left to remove it, is to patch the source of Konsole.

Good luck.