I have a simple image with an off-white background. I would like to make that off-white background be pure white. I can use the "Replace Color" tool to select the off-white but I don't know how to then get that selected color to be set to my desired color of pure white.
I can recommend the use of the magic wand tool¹ to select only the white you wish to recolour using the tolerance meter. Afterwards it is a simple matter of using the paint bucket tool set to the desired color of pure white.
In CMYK mode go to Curves and pull the lefthand slider along. Check the off white areas (now hopefully white) with the eye-dropper tool and keep dragging along until the eye-dropper registers 0 colours. You shouldn't need to drag it far (a few points) and beware as it will alter some other colours slightly.