It was my understanding that a past Covid positive test would be listed as one of the factors in reporting deaths to the DOH for their status reports on Covid numbers.
I don’t know whether a past positive test would mean reporting the the virus as a significant condition when certifying the death as stated in CDC guidelines.
The death certificate form has multiple locations for listing different bit of information. Things like the immediate cause of death, the underlying cause of death, and so on. If COVID-19 was present, but not part of the sequence of events leading directly to death, it would be listed in Part II.