The real question is, do you want to "feature" the door by making it stand out, which is what would happen if you stain it. Featuring it is something you want to do to entice people to walk into the adjoining space. If your intended stain color is medium to dark, staining can make your room look smaller. With the door already painted, you will have to strip the paint before you stain and varnish/polyurethane it. If the door wood has a visible grain, such as some pine and fir, you might want to coat the stripped wood with a sanding sealer or other sealer to "close" the grain before you stain it to match or contrast with your ash floor.
Try to visualize what the door will look like stained if all the other doors in the room are painted, or do you want to strip and stain them as well.