Can zinc capped and stainless capped screw fasteners corrode?

Yes. While they are very resistant to corrosion they can corrode.

Regarding fastener heads, yes. While the bare zinc cap and the bare stainless cap are very corrosion resistant, they can corrode (via pitting of the surface) in extremely corrosive (direct seacoast, direct heavy industrial) environments. Painting the heads of the fasteners enhances performance by providing a barrier coat to the metal.

Regarding fastener shanks (from point to below washer), yes. In this case, the fastener shanks are very susceptible to corrosion if the sealing washer leaks and/or there are corrosive elements subjected to the underside of the fastener. The shanks of these fasteners are zinc plated and/or organically coated carbon steel that will red rust, and if severe enough, negatively impact performance of the fastener.