If you wanted to run 31 inch tires on a Toyota Tacoma would you want to put them on 15" or 16" rims?

I would do 16 inch as some 15 inch wheels do not clear the calipers. However most 16 inch tires come in metric sizing so you won't be able to get one that actually says 31 on the sidewall. But you can of course get a metric tire that is approx. 31 inches tall.

You will probably need a lift kit or something else, fender flares... body work etc to make them fit

Body work for 31 inch? Not needed. I've seen lots of people fit 35's while stock. They do have rubbing issues though. 33 inch is a good number. I have 33's and a 3 inch lift (though I believe they will still fit without a lift) and only rub slightly on the frame when tightly turning. If I'm flexed out in just the right way they will also rub on the bolt head of my rear left fender flare. Really...taking away the flares would help with rubbing, not adding them on.

A 31" tire will fit without changing anything. I have a 2003 prerunner with 265/70r/16 which is a 31" tall tire. they will fit on a stock rim and protect the rim beads