A dead body can stay in a morgue for as long as it's needed and remain in the freezer until the body can be prepared and buried.
Usually though after a body is put into a morgue and body freezer they remain there for between 1 to 3 days at most before the funeral home comes and gets the body to prepare it for embalming and prepare it for the funeral.
Then the body can sometimes be kept at the funeral home for sometimes as long as a week or 2 while out of state family members arrive for the funeral.
Most bodies are either cremated or buried within 2 to 4 days of their death though so most bodies don't stay in the morgue very long or in the funeral home very long.