It is it possible to safely use a 110V variac (variable transformer) at 230V?

No, the core will saturate and (if you're lucky) blow a fuse.

If your 230VAC happens to be (say) 400Hz, the answer might be different.

Wikipedia has a very nice animated image showing the effect of too low frequency. Too high voltage has the same effect. The value of (maximum primary voltage)/frequency is a constant for a given transformer design).

As you can see, the magnetizing current spikes up twice per cycle and the resistive heating losses increase with the current squared.

While Sphero's answer rules if the device really is 110VAC input only, my experience with Variacs is that most of them are 240V friendly - to the point that they come packed with a dial that can be flipped to read correctly for 120VAC in or 240VAC in - so you might want to look "inside the case" and see if the Variac itself is indeed only 110VAC labeled, or of it's a 100-240VAC variac in a case that "makes it" 110VAC due to connections and external labelling.

We used a lot of them to control high voltage supply outputs by controlling the input voltage of the supply back in the 1980s, so I've got some practical experience with them.