Will a full frame lens work on a non-full frame camera?

I have a Nikon d7000 and I'm interested in getting the Nikon 14-24mm

Absolutely. The only difference with using that lens on a DX body like the D7000 is going to be that you have to take the crop factor into consideration. The 14-24mm is going to give you an angle equivalent to a 21-36mm lens on any of Nikon's DX bodies.

Like the first answer said, it'll work. However a lot of people consider it a bit of a waste to put a lens like the 14-24 on a crop sensor. You don't get to use all the lens, and given that this lens is designed specifically for very wide angle use… Plus, there are good wide angle equivalents for the crop sensor which will get you pretty close to 14mm, and unless you're aiming for the highest possible image quality, then they're worth a look as well.

Plus the 14-24 is a big lens and very expensive. You can't put filters on it because the front element sticks out, and it could be easy to damage if you're not careful.

But it's your money and choice in the end! I'd suggest going to a co-operative camera shop and trying out the 14-24 and DX equivalents as well; you may not see enough of a difference in the end picture to justify the 14-24.

Akuseru is right. If you plan to change to full-frame in the future get that lens. If you will stick with crop-frame think of a wider lens if you really want width. That lens works as a 21 to 36mm on crop-frame.