That depends on what the poison is and how it is stored.
Many oganic poisons like most common insecticides and herbicides break down over time and eventually lose their toxicity. They become less poisonous, meaning you need a bigger dose, but you'd have to wait years after the expiration before it stopped working altogether.
Some organic poisons have an expiry date for excatly the opposite reason. They break down, but they break down into more poisonous substances. Some herbicides for example gradually decay and produce dioxanes, some pesticides break down when exposed to water and produce a very potent volatile nerve gas. You can usually spot these poisons beause the expiration date is very short and is printed as two dates: expiration date and shelf life from date of opening.
The inorganic poisons like arsenic or mercury compounds tend not to break down as such. They have expiration dates because they absorb moisture from the air and change form chemically. That makes then harder to use and it also means they take longer to work. Instead of killing the animal within hours it might take days or months. Ironically the chemically altered form of arsenic is often more deadly insofar as it requires a smaller dose to be lethal. It's just less useful because it takes a long time to act.
So the asnwer is, it can become less poisonous, it can become more posionous, but poison is always poison.

It becomes less poisonous