This is usually the heating element - corrosion lets small amounts of water through the element jacket, making the mineral insulation inside slightly damp. When the machine applies power to the element, enough current leaks from live to earth to trip the RCD.
In the absence of a proper electrician's insulation tester (a DVM in ohms mode doesn't test at a high enough voltage to give a meaningful result) the best way to prove this fault is to (carefully, and with the power off!) disconnect and tie back the wires that go to the element, then run the machine and see if it gets any further in the cycle.
A.
marko, May 2007