No, It Won't On It's Own
What phosphates will do is deal with mineral/hard water residue, which when cleared makes a surface appear cleaner.
Case in point, the stainless steel wash tub on my Miele shines like new, and the frequent use of phosphate laden or added detergent is the cause IMHO.
Certain stains or their remaining traces can only be "removed" with bleaches. Used the words in quotes because bleaching normally does not remove stains, but merely breaks apart whatever colour stains have.
If you take a white item that has been stained and washed with bleaching agents, hold it up to the light and you will most always see the outline of the stains still there. You cannot see the stain because it's colour has been removed, but never the less the stain is there.
Chlorine bleach does not care what colour it removes. Dyes, fruits, veggies, blood, etc all often can be decolourised by the stuff. Oxygen bleach OTHO acts on natural "dyes" such as tannins and such,this is why they are called "colour safe" bleaches. Long as the dye used on a textile is colourfast and one does not subject the item to very hot to boiling water temperatures, the colour should remain. Though frequent use of oxygen bleaches, especially activated and or percarbonate bleaches can and often will fade colours. This is why Persil and other European detergents have special versions for colours, formulated without bleaches and often bluing agents.