Does GTA V's main campaign have full coop play?

The main campaign is single player only but there is to be co-op play online. Multiplayer is meant to be a persistent world where players do take on co-op missions (as well as the usual array of MP game modes).

Watch the multiplayer teaser video to see what's included online...

No.

I was just able to find the answer at GTA V: Burning Questions Answered:

Why a single-player story with multiple characters? Why no multiplayer co-op?

It would be impossible to do that and keep the level of precision we've got in this. You could make a great co-op game, but we felt that we're doing other bits of our multiplayer that will fulfill those desires in ways we think are very fun.