As my car is now 6 years old, it has accumulated a lot of dead skin on the door armrest. When it's humid outside, my arm tends to stick to the hard plastic, making a most unpleasant feeling. How does one go about removing this layer of dirt? I have tried soapy water, a microfiber towel and a magic eraser but it still manages to stay dirty. Anyone have any tips of how to get rid of this issue?

 

Layer of fat and other residues from skin contact require stronger cleaners. I would try mineral spirits and denatured spirit i.e. ethyl alcohol based cleaner.

 

You might try heat/steam cleaning.
Check out some of Larry Kosilla's (AMMO NYC) detailing videos on Youtube for ideas and pointers.