There's a ton of money in the Gulf to fund anything Gulf Arabs like and Gulf Arabs love fighting against Shiite infidels, so yeah, they'll fund any amount of Sunni rebellions with tacit backing from the US as it gives them a chance to paint the current regime as anti-democratic. In Syria right now there are tens of thousands of Iraq war reenactors who'd join in too. In a decade or three you may see a Greater Syria containing the Sunni triangle, an independent Kurdish region extending into other countries and a Shiastan from Bagdad southwards that contains all the oil (apart from the Kurdish stuff.)
I read a pretty good article on growing Kurdish independence recently. I'm not sure if the gulf states have the desire or juice to make a Sunni insurgency relevant again. I'd have to believe that Iran would double down their efforts particularly with Assad taking his licks. If this turned into a real civil war it would be interesting to see if the Kurds try to go 100% independent.