Not necessarily.

Aspirin yes, paracetamol no.

Some medicine also just becomes less effective, not more dangerous.

In the case of tetracycline antibiotics the degradation products can damage the kidneys and cause Fanconi syndrome. So in that case as a medicine for people it becomes poison, as a poison for bacteria it becomes safer.

If you wait long enough does poison expire and become medicine again tho?

Someone hire this guy! Nobel material.