How long does bad breath associated with nasal foreign body last?

Until Infection Heal: Metallic taste from a nasal foreign object is either from the object chemically reacting with the nasal mucosa or causing an infection. Normal taste should return in 3-7 after the object is removed and the local infection is healed.

Very short-lived : It should not last very long after removal, unless you've developed a sinus infection from it. Try a neti-pot or sinus rinses twice a day for 3 days, if not gone by then go see a nose and sinus specialist.