Does Chalice of the Void counter spells cast with alternate costs if their converted mana cost is 0?

Chalice of the Void counters a spell when it is cast if that spell's converted mana cost (mana value) equals the number of charge counters on Chalice. Crucially, this check uses the spell's mana cost as printed in the top-right corner of the card — not the cost actually paid to cast it.

Under CR 202.3, a spell's mana value is calculated from its mana cost (the printed cost). Alternate costs, such as Flashback, Suspend's free cast, or paying Phyrexian mana, change what you pay but do not change the spell's mana value (CR 601.2f, 601.2b). So a spell with a printed mana cost of {0} — like Ancestral Vision (which has no mana cost at all, giving it a mana value of 0 once on the stack via Suspend) or a spell cast for free via Force of Will's alternate cost — can still be countered.

A special note: spells with no mana cost (like Ancestral Vision or Evermorn's Cascade targets) also have mana value 0 (CR 202.3a), so Chalice with 0 counters counters those too.

Concrete example: Your opponent has Chalice of the Void with 0 charge counters. You cast Gut Shot by paying its Phyrexian mana alternate cost of 2 life instead of {R/P}. Gut Shot's printed mana cost is {R/P}, which has a mana value of 1 — so Chalice does not counter it. But if you cast Memnite (mana cost {0}, mana value 0) for free via some effect, Chalice with 0 counters will trigger and counter it.

Bottom line: always check the printed mana cost's mana value, never the cost paid. Chalice's trigger (CR 509.2 for triggered abilities going on the stack) fires on cast, and the counter happens when the trigger resolves, regardless of how the spell was paid for.