Should my unfinished siding be left uncovered when it rains?
I think you're asking about sheathing, not siding. What we see above the brick are oriented strand board (OSB) sheathing panels. They should not be left exposed to the weather for more than a few weeks (or a few rain events, whichever comes first). 

Ideally, housewrap and other components of the drain plane are installed immediately to protect the sheathing and other weather-sensitive parts of your home.