Does superman's heat vision come from his eyeballs or eyes? If he doesn't move his head but only moves his eyes, will his heat vision move?

Eyeballs are eyes. And yes, the heat vision comes out of the eyes.

So it follows his eyes? Remember the scene in MoS? Why didn't Zod just look at the people?

It does.

Plot would be my guess but some people could argue that Zod just was trying to force Superman to kill him, since his whole race was dead for good.

I knew Zod was good after all!

He's not at all. He slaughtered thousands of humans. Maybe tens of thousands.

Eyeballs are part of the eye. It actually makes sense, because heat vision doesn't just come from the eyeball, it completely engulfs the entire eye sclera included, so just looking wouldn't change it's course, Zod would need to actually move his head.

The sclera is part of the eyeball, which makes up the eye. I'm an anatomist, you aren't going to snow me on human anatomy.

The heat vision was never shown to engulf the entire eye (as most of the eye cannot be seen without dissection) and the comics have shown very specifically that the heat vision comes through the iris. The movies have no reason to change it and it would just look stupid to see him bapping his head about to do something like writing with his heat vision.

Considering the nature of heat vision and how it essentially translates to "shooting light" (I think that's it. Something about shooting solar energy from his cells), it would make sense for it to come out of the iris.

Also, it should be the iris. It shot where he was staring, not just where his head was. As for Zod... maybe Zod just wasn't very bright. But it's quite possible that he wanted to die at that point.

If it did come through the iris, then why is heat vision always in a completely straight line? Irises move, so I believe even though it looks like it's coming from the iris, it is actually the position of the head that aims it. Make sense?

It's a straight line because it's a beam, not a spray. An iris only becomes dilated or constricted, there is no iris I am aware of that can move in any other way. The iris is not a body part so much as it is an absence of a body part. It's literally a hole in the eye.

In the movie both superman and Zod had to move their head to essentially aim the heat vision. I'm not sure if this was because of not mastering the art, or because that's just the way it is. I think that it was because of the former in my opinion.

Considering that most Images show him firing straight, I would say that his heat vision comes from his eyes and it can be controlled by moving his head.

Maybe Superman only turns his head while shooting his heat vision because he's unsure if he looks down/up/to the side enough just by moving his eyes it'll burn his own skin (eyelids, mostly). Anyone ever think of that? The moving of the head is just a safety precaution.