We have just bought a 1930's house and it has already varnished floorboards but we want to darken the colour. Will we have to sand back and start from scratch or can we varnish on varnish?
You might try a glaze coat and then over-seal with more varnish.
A pigmented stain is made from 'bits' of color (the pigment). When applied to bare wood, most of the color is on the surface. (But HerrBag, I hear you saying, my Minwax stain is oil based and penetrates. Alas, Minwax is a MIX of dye and pigmented stain.
Most paint is pigment mixed into a base material that lets you disperse it evenly. The opaqueness comes from the concentration of pigment and the base material itself.
are the same pigment in a transparent base that lets you 'float' a color layer over previously finished wood. Indeed, to make a custom glaze, you add a small percentage of regular paint to the glazing base.
To adapt the technique for large areas will require the slower setting version glaze and a swivel head applicator (as used by flooring pros, its an oversized tubular paint pad (like a paint roller that doesn't rotate)).
A good article on glazes and glazing from Woodworkers Journal is here.
You have to get the poly or any protective coating off first. If you floor doesn't need additional sanding then you do not have to sand off the varnish layer especially when going darker. I suggest you take the poly off in a small area and see how a few different colors will blend with whatever was first applied.