Here's the thing: in basic aqueous solution iodine forms hypoiodite initially. The hypoiodite then rapidly disproportionates to iodide and iodate.
Iodide probably just sits around and doesn't do much, but iodate is a very powerful oxidizer, and methanol probably gets oxidized to formaldehyde and formic acid. Since you have NaOH you could form sodium formate, which *might* precipitate if you have a large excess of methanol compared to water.
Actually why don't you run some tests on the precipitate? NMR and IR should give you some clues as to whether it's an organic or inorganic precipitate. I have a hunch the precipitate is NOT iodoform, and probably something inorganic.
Also triiodomethanol would be unstable, its sodium salt much more so. An iodide would simply be kicked out to form carbonyl iodide, the iodine analogue of phosgene. This would be supremely reactive and react either with water to give carbonic acid which spontaneously decarboxylates, or react with methanol to give dimethyl carbonate. Of course neither of the two are solids, so...