gerald_clark: I'm sure it isn't a bios issue because the machine will boot into the bios setup screens (not to mention DOS, ttylinux, damn small linux, and several others) without a monitor attached. Only booting TC hangs.
rich: we don't have a full time hardware person right now, so I'll need to wait until he's back in the office to try building a special cable. However, it has always been my experience (in the past) that anything written to a text mode display is in memory and will be displayed as soon as a monitor is connected. When I connect a monitor to a hung TC, the screen is blank, except for the hardware cursor.
tinypoodle: in order to follow your suggestion, I must build my own TC boot image. That would just open a whole new can of worms by allowing everyone here to blame my build procedure, settings, environment, whatever for the problem. No, I'm working with clean TC ISO images, downloaded directly from links on tinycorelinux.net, burned to CD and booted in one of our embedded boxes with an IDE CDROM as the only attached disk drive.
EVERY other OS that I've put on CD and tried to boot without a monitor has worked. TC hangs. I was even able to boot a Ubuntu 12 installer CD I had up to the first GRAPHICAL screen in the installer -- it did not hang -- and when I connected a monitor, the graphical screen was presented!