I have wired a cooktop to a 30 amp breaker but the specs call for a 40 amp. What will happen?

Do you mean apart from the electrical code violations?

If the cooktop tries to draw the full 40 amps, the 30 amp breaker will trip.

If the specs call for a 40A circuit why did you wire it to 30A?

It will trip the breaker and you will not be able to use the unit.

Can you tell us how this happened? Was the wire already in place? Did you simply make a mistake? It's not going to hurt the appliance, but it will limit just how much of it you can use at once. The breaker tripping will get annoying enough for you to do it correctly.

have you tried it?

I'm guessing they were trying to avoid spending the roughly $12 for the correct breaker or didn't find out what the requirements were before wiring the job and undersized everything.

Don't change that breaker unless you also change the wiring.

Agreed... what size wire is in it now?