Does extort trigger if you cast a spell with a mana cost of zero?

Extort triggers whenever you cast a spell — full stop. The triggered ability reads: 'Whenever you cast a spell, you may pay {W/B}. If you do, each opponent loses 1 life and you gain that much life.' Nothing in the extort ability checks the mana cost of the spell you cast, so a zero-mana spell like Ornithopter or Gitaxian Probe (when paid with life) will trigger it just as well as any other spell.

Per CR 702.101a, extort's trigger condition is simply casting a spell. CR 601.2 defines casting a spell as the process of taking a card from where it is and putting it on the stack, which happens regardless of the spell's converted mana cost (now called mana value). A zero mana value does not prevent the spell from being cast or from causing triggered abilities to trigger.

The only requirement for extort to trigger is that you are the one casting the spell and you control a permanent with extort. Once the spell is on the stack and casting is complete (CR 601.2i), extort triggers and goes on the stack above it, giving you the opportunity to pay {W/B}.

Concrete example: You control Syndic of Tithes and cast Ornithopter (mana cost {0}). Extort triggers. You choose to pay {W/B}; each opponent loses 1 life and you gain 1 life per opponent. Ornithopter then resolves normally.

Note: If you control multiple permanents with extort, each one triggers separately, giving you multiple opportunities to pay {W/B} and drain opponents for each trigger (CR 702.101b).