What foreign nationality(ies) you have and whether you have passport(s) from those countries are not relevant to your US permanent resident status.
Most permanent residents have passports from their country(ies) of nationality; that is the normal way that permanent residents travel abroad (as they cannot hold US passports). A few permanent residents don't have passports, maybe because they don't need to travel abroad, or they are stateless or fear their country of nationality or otherwise unable to get a passport from their country of nationality in which case they may travel with a Re-entry Permit or Refugee Travel Document. Either way is fine as far as the US is concerned.
The US doesn't actually require that permanent residents have a passport to enter the US; but practically no airline will board you on an international flight without a passport or equivalent travel document (except maybe to Canada).
The only situation in which getting a foreign passport is potentially problematic is if you are an asylee, in which case getting a passport from your country of nationality can be viewed as availing yourself of that country's protection again, so you don't really need asylum anymore. But if you are a permanent resident, it is not a problem.