If you use Color Remover, it will most likely not remove the color from a print or logo. Accordingly, the color you choose to overdye with will mix with the existing color(s) in the fabric and create new colors. In other words, the rules of color mixing apply.
When selecting a color, think about how it will mix with the colors in the pattern. For example, if you were to use a blue dye on a white item with a pink flower pattern, the pink flowers will turn purplish (blue and pink make purple) while the rest of the fabric will dye blue. Even if you were to overdye the patterned garment with a dark color, you may find that the color will appear lighter along the patterned area.