You find diverging opinions because it depends.
For laser (so toner) presses, 1200 dpi printers will consume less: they pile fewer toner particles on top of each other, and they allow for a higher lineature in the AM screen. The latter allows for white space to fill up faster, helping to reach higher densities with less toner.
For inkjet presses, the story is more complicated. Inkjet presses use an FM screen, and how that screen is built impacts ink consumption. If the print head allows for several drop sizes, most screens fill the images with the lowest drop size first, before adding larger drops. That helps getting rid of white space quickly and gives lower ink consumption. But 1200 dpi heads do not always have several drop sizes. In that case, the 600 dpi head will use less ink in lighter images. For heavy color images, the 1200 dpi will always use less, because the drop is smaller, also less high and this lower ink volume.