"This bird doesn't chirp."
It depends on the context, specifically the subject, typically if you're talking about an animal, it'll be 鳴く and a human or something crying based on an emotional response (maybe any sentient being? not sure on that specifically), it would be 泣く.
For example: カエルが鳴いている。 (The frog is croaking)
女の子が泣いている。 (The girl is crying)