I was once asked to look at a heavily used microwave oven at a local residential facility. The description was similar, but the problem was obvious on first inspection. Long finger nails punching through the elastometric contact keyboard. Once the protective membrane on the front panel is punctured, all manner of liquids and gasses can cause damage. I've seen this problem other times, where the customer doesn't want to admit that they damaged the front panel, or somehow considers such damage as normal. It also applies to TV/hi-fi remote controls, printer control panels, calculators, where easily damaged flexible membranes are used in place of hard plastic buttons.
While grease is usually non-conductive, having any grease in the various connectors will produce lousy connections. I suspect a good cleaning, and a low grease diet, might restore the oven to normal operation. The green crud (copper sulfate) does not necessarily signal the demise of the wiring, unless it's extensive. I would not replace any electronics until the keyboard and controller cards are baptized in solvent. It would also be interesting to determine how the grease entered the area. Probably a mechanically mis-aligned exhaust duct.
Opinions:
This one is interesting:
The Bottom Line: Despite the top-flight brand name, attractive styling, and high price tag, this microwave is unreliable and extremely difficult to clean.
Drivel: US DoE can't figure out how to measure microwave oven power output.
Sigh.