You should most definitely make sure to refrigerate the tuna that you get home from the supermarket, whether it is canned or fresh, in order to avoid it going bad. Since tuna doesn’t stay fresh for a very long time, you will need to keep it in a sealed container in the refrigerator until you are ready to eat it.
By simply leaving your tuna out of the fridge, you are risking bacteria growing on it and when it is ingested it could cause food poisoning. It will also be important to make sure that you do not thaw out and refreeze tuna as bacteria can also grow on it this way. Instead it will be important to keep your tuna frozen or refrigerated until you want to eat it.