Can I use a 240v variac with 120v?
You will need to ensure that you do not exceed the maximum current as well as the maximum VA rating.  

For example, if you have 2400 VA variac designed to work at 240 V, then the implied maximum current rating is 10 A.  You can't safely exceed this amount. The maximum VA rating ends up being proportional to the voltage dialed in on the variac (but never more than 2400 VA):

$$ VA_{max} = (10 A) (V_{output})/(V_{mains}) $$

The reason for this is that the power dissipated in a conductor is proportional to the square of the current.  If you exceed the current rating the Variac may overheat regardless of voltage.

Also, if you are using a 50 Hz variac with 60 Hz mains, then the autotransformer will be somewhat oversized, and you have a bit of a safety margin.  If the situation were reversed, then the autotransformer would be undersized, and you would need to further reduce the maximum current.  I believe it's proportional to 50/60 in this case, but I can't remember the formula for this.

I assumed based on the 120V/240V that you're working with a normal, residential sized loads.  If for some reason this is an industrial load, you'd better just use the right equipment for the job.  Either way, test it first on a low power load.