That would be very bad for all parties involved. We don't want Iran openly supplying *expletive deleted*it militias. They slip them more than enough material at the moment. But if they opened the logistic pipeline, things could get touchy.
I guess I'm not convinced they aren't pretty much doing that already, and we're just turning a blind eye to it. I have nothing to back that up of course, it's just my thoughts on the subject.
Oh, Iran is supplying the Shiites with arms and more importantly, technical expertise. You might have noticed occassional sudden bursts of sophistication in the IED's in the *expletive deleted*it areas. Shaped charge IED's are getting old hat. Explosively formed penetrators (EFP's) are the interesting development.
We're not exactly turning a blind eye to it as much as there's not much we can effectively do about it. We can run interdiction to cut down on the amount of supplies, but we can't stop it short of invasion. For ethnic solidarity reasons, Iran is almost obligated to supply the *expletive deleted*it militias to some extent. To not do so would be rather baffling. But there's a difference between slipping SOME supplies as a face gesture and opening the floodgates. Iran domestically manufactures RPG-29's in large numbers. Imagine if they slipped hundreds or thousands across the border instead of a few dozen.
Iran is probably gauging a lot of its actions on a long term goal of incorporating *expletive deleted*it Iraqi territory into Iran proper. If we were smarter, we'd be building the infrastructure of the Kurds at a much higher rate and arming/training the Kurdish militias like no tomorrow. Specifically, teaching them to run a guerilla war against Iran if required.