Simple, really. To unfog the green mist, one only has to look at fog in general. Yes, it obscures vision, and it swallows up objects that enter into it. This is a spiritual fog, which affects humans, in particular, not necessarily inanimate objects like the Dawn Treader, itself, as distinct from its crew and passengers.

No of course we know it doesn`t make sense...
In the end we can come up with all sorts of justifications and reasons for these things, but if they weren`t presented in the movie, then there is no point. The movie should give us these answers.

I understand suspension of belief, but there is a difference between that and a lack of logic. We can have magic in a movie, but it has to follow the rules set up within the story.

The "multitasking" mist...
I was also confused by the green mist and its actions. Actually, it's a quite good question why the DT isn't "eaten" by the mist. I can't really think of a proper answer, but it somehow makes sense that the DT is too big for the green mist to transport to Dark Island.

By the way, I can't really see what threat the green mist is for the crew. Of course the mist is able to transport people to Dark Island and to tempt the crew, especially Edmund, Lucy and Caspian... But what else is the mist able to do? Is it possible that the mist kills people? I mean, the mist itself doesn't seem dangerous to me at all (compared to e.g. the White Witch in LWW). What would have happened if Ed had given in at the end? Probably, the green mist in the form of the White Witch wouldn't have killed him. The sea serpent... yeah maybe, but the sea serpent is a creation of the green mist just like the imagination of the White Witch, isn't it? So what makes the sea serpent more dangerous? It is also just Edmund's imagination...

So many questions... Just tell me if my trains of thought are too freaky

But what else is the mist able to do?

Good question! How do we unfog the fog? Mostly the green mist does foggy things, like seep everywhere, even on board. Just as well Edmund had that torch still, which also might have protected the Dawn Treader . Though fog lights and head lamps might have been better. Otherwise I suspect that the green mist was just a metaphor for a spiritual fog and that its inventors hadn't the foggiest idea to what else the villainous fog could do, apart from the usual accidents, illness and depression fogs do sometimes induce in real life.

Of course this is no ordinary fog. It is a poisonous fog emanating from Nightmare Island AKA Dark Island. It just lurvves bad thoughts and horrible deeds.