I know in theory the reaction of methanol with hydrogen iodide will form methyl iodide and water. This is part of the Monsanto process for acetic acid. But the problem is that back reaction will also occur.
Also if you try to use concentrated sulfuric acid to stop the back reaction (water plus methyl iodide to form methanol and HI) being a problem. But then you will form HI from the KI, the HI tends to decompose into hydrogen and elemental iodine. This will make the reaction very messy.
The Dean-Stark head will be useless for this reaction as methanol (methyl alcohol) and water are totally misicble. A better dehydration reagent would be trimethyl orthoformate. If you were to mix methanol, trimethyl orthoformate, sodium iodide and sulfuric acid I think you would have a better chance of forming methyl iodide. But the purity would not be good as it would have methyl formate mixed with it.
The easy way to make methyl iodide is to react a solution of sodium iodide in water with dimethyl sulfate. If you put the sodium iodide in a pear shaped 50 ml flask, add a stir bar and then the dimethyl sulfate you can then heat it up and the methyl iodide will distill out as it forms. I think that this reaction has much more promise for a methyl iodide synthesis.
Dimethyl sulfate can be formed by the reaction of sulfur trioxide and methanol, so in industry it is a good and cheap methylation reagent. This could then be reacted with sodium or potassium iodide to form the methyl iodide.
I tend to put a still head on top of a 1 inch long fractionation column and then use a liebig condensor to collect the methyl iodide. The only reason a chemist does this is if you want to make I-131 containing methyl iodide. I have found that if you add I-131 as sodium iodide to the synthesis before you add the dimethyl sulfate that you can make radioactive methyl iodide.
When I do this I work in a very good fume hood, while wearing nitrile gloves to handle every object. It is important with this reaction to pay attention to both the radioactive and the non radioactive hazards. The dimethyl sulfate is a very nasty toxic reagent (carcinogenic). I tend at the end of the reaction to allow it to cool. I then add some diethylamine to the still pot and then distill the diethylamine through the glassware (by this time the methyl iodide has been placed inside a glass vial over copper wire). The idea of the diethylamine is to make sure that no dimethyl sulfate remains in the glass equipment.
I then allow it to cool, I then wash out all the glass ware into a plastic waste bottle. I then put the glass equipment inside plastic bags. The glass is allowed to stand in a fume hood for weeks while the radioactivity decays away. The waste is allowed to stand. I have always used I-131 while has a half life of 8 days, but I am sure the work could be done with a longer lived radioiodine.