Will leavin the lens cap off my DSLR for 3+ weeks damage the lens?

I just found my Nikon D3000 in my sisters room with the lens COVERED in dust and when I looked for the cap it was CAKED in dry nail polish. I'm soaking the cup in nail polish remover but will I be able to use it again? And is my lens going to be damaged at all? I'm so mad!

Your lens should be totally fine, that is, if she didn't scratch it. Use your lens and you should be able to tell right away.
It would also be a great idea to clean your lens very thoroughly with a dry cloth. If that doesn't get all the dust off you can use a lightly damped cloth(be careful though!).
Make sure your lens cap is completely dry before using.

And as for that sister… I would have killed her.

Soaking your plastic lens cap in nail polish remover will do more damage than the nail polish

Better to just spend the few dollars and buy a replacement and 1) have your sister pay for it and 2) NEVER let her near your camera again

You do need to place a UV filter on your lens to protect the lens from becoming damaged by uncontrolled events

Are you talking about damage to the lens or to the lens cap? (your title says lens but the main paragraph of your question says lens cap).

Your lens cap may be ruined, but your lens sounds find. Just clean it with some lens tissue paper and/or a blower brush.

Or better yet, why not just take a picture and find out?