I got something called stinging vial from the donation corner. What does it do if I drink it?

it will eventually make you temporarily sick enough that NPC Zosi will not be able to tell you are not a zombie. He will allow you to make Alchove combinations in his shop until you start to heal and get better.

If you drink it during Survival, it will also grant you a Skin.

If you drink it right now, it will make you progress through the stinging virus and you'll have that skin temporarily.

Or you can save it for Survival next February, and if you drink it during that event you'll get to unlock and keep the skin.

If I use it, can I visit the zombie den?

you do not need to use anything to visit the Zombie Den. Skitters opened the Zombie Den to the living-public three years ago.