Oh...
The output of the PID is normally a DC voltage that is fed to the DC side of the SSR.
http://www.ielogical.com/coffee/SSRMount.JPGThe RED wire, connected to +, is the one you need to lift.
If you remove the control input from the SSR, the machine should stay cool. If the SSR is blown closed, an unusual failure, the machine will heat regardless.
Is it possible that you have the SSR control wired to the wrong terminal and it's always on?