Do fence pickets need to be totally dry before installing?
That depends partly on the fence design. If you're doing something edge-to-edge, installing water-swollen pickets will result in gaps once the wood dries out. 

Otherwise, it's more likely that the pickets will warp if left to dry quickly in the sun. The side that dries out first contracts, causing curvature. 

That said, pressure-treated lumber is extremely wet to begin with, so it may not matter much. If you get any severely warped pickets you might just have to change them out. 

I'd buy an inexpensive tarp and keep it all dry until it's up. Why make the project more challenging than it already is?