If I am renting a house and our heater stopped working, are we responsible for the repair bill?

In California, the landlord has a duty to maintain and repair the premises and provide a habitable residence for tenants.  Heating obviously falls under the umbrella of habitable premises. That cost is almost always on the landlord, not the tenant, alhough you should check your lease agreement for any contrary language.

For additional reference, check out Calif. Civ. Code Section 1941 et seq.