I misread what you stated, but it's not really dirt, but rather just bit-rot that just builds up in the unit from continual use and age.
If I had to make a guess, the first thing I'd check for are bulging/leaking capacitors. Sometimes the electrolyte inside a capacitor goes bad and tries to push its way upward and eventually starts leaking through the top.
At work, I replaced two video cards not too long ago in some high end Dell's whose capacitors did exactly that.
Just a theory anyway as I've never had the opportunity to see a Star 4000 up close.