Methyl Iodide synthesis
So what I said in the comment still holds: it could work (particularly in combination with a Dean-Stark apparatus to remove excess water), but there is no way to know for sure unless someone tries it out.

The more usual (and early) method is described in Organic Syntheses and utilizes the reaction of methanol with iodine in the presence of phosphorous



I don't know the exact reaction mechanism (if anyone has a clue, please feel free to edit!) but what happens is that $\ce{PI_3}$ is generated in situ from the elemental iodine and phosphorous and reacts with methanol to yield the methyl iodide and phosphorous acid

$\ce{3 CH3OH + PI3 -&gt; 3 CH3I + H3PO3}$

Phosphorous acid does not act as nucleophile and so the problem of a competing backward reaction, which you would have in your proposed synthesis, is largely prevented.

I'd also like to add some caution: the reaction described in Organic Syntheses is not suited for beginners or in amateur setting. The chemicals involved are nasty: highly toxic, highly flammable, alkylating and pyrophoric. Don't try this at home :)