To cover orange hair, use bleach to lighten the hair, or a gloss or darker shade to cover it. When hair is dyed to a lighter color, like blonde, it may look brassy or orange, particularly when using a store-bought hair coloring product. Those products don't have the strength of professional brands to lighten the hair properly, especially if the person's hair isn't already a light shade.
When dying the hair to a lighter color, the chemicals strip away natural molecules. This is when chemical damage is likely to happen. Darkening the hair, on the other hand, is much safer because instead of stripping hair molecules away, the dye is simply depositing artificial colors into the hair.
Besides applying bleach, there are other options available to those who need to fix orange hair. A professional stylist can apply a blue or violet-based toner to the hair to lighten it out of the orange color. The toner does this by acting as a semi-permanent color glaze, and since blue and violet are opposite colors from orange, the glaze "cancels out" the orange color.
Simply apply a darker permanent coloring on top of the orange to hide it. Brown shades are the easiest option in this case, simply because brown shades are predictable and easy to control in the coloring process. Brown colors with cool tones, like blue, green and violet, help to neutralize the orange. Using a warmer brown will create a more orange-brown shade. It's recommended to stay away from store-bought products and employ a professional hair stylist, especially when trying to dye the hair to a very different shade.