4K might be useful for a projector if you are using software trapezoid correction. I have a 1080p projector, but mounted it above the vertical centre of the screen, so I have to use the in-built trapezoid correction to make the picture rectangular. This means that only one line in the display had 1920 pixels, and the remaining 1079 lines each have some interpolation.

If I had a 4k projector, even with a 1080p source, the quality of the projection would presumably be higher.