Mains AC overvoltage protection using a TRIAC
However, when I try to simulate it in Multisim, the TRIAC remains off
  no matter what the input voltage is. I think the transistor switches
  on and off properly though.


The signal that attempts to activate the triac (Q1 emitter) is referenced to a floating supply and isolated from the triac (in all other respects) by the transformer. 



This means it can never push/pull current through the triac's gate.

I'm not ruling out other problems too but, once you have this corrected your sim should indicate other potential issues.