Should I repaint this dryer part?
I don't know much about combination washer / dryers, but are you sure you've identified the problem correctly? Why would there be so much soap residue in the dryer exhaust. I don't recall ever seeing any soap residue on a dryer. And why is there a heating element in the on the exhaust side of dryer?

To answer your question, though: if this part is subject to high heat, which it sounds like it is, I would use an oil paint designed for radiators, or some high-temperature Rustoleum.