When is it necessary to paint a wooden fascia (barge) board?
Exposed wood must always be sealed in some fashion unless it's chemically treated or of a species that's rot-resistant (cedar, teak, etc.). Even then it tends to preserve the aesthetics of wood to paint, oil, or varnish it.

Metal fascia is commonly installed right over "subfascia" (raw SPF 2x6 lumber). 
If everything is installed correctly it should never see water.