I'm thinking of something similar to the liquid used in the classic dipping birds. Temperature differential in two "bulbs" at each end of a tube should cause vapor to expand and push liquid to the top, shifting the weight and tipping it. The closer to room temperature the boiling point is, the better. Thanks.
Yes, there are lots. Start with Wikipedia's list of refrigerants; while it doesn't call out flammability, any fully-halogenated compounds will be effectively non-flammable. Trichlorofluoromethane has a standard boiling point just above standard room temperature.
You didn't say anything about pressure in your question. If you don't mind pressures higher than one standard atmosphere, and don't mind toxicity, candidates like ammonia, sulfur dioxide, and chlorine come into play. If you don't mind low pressure, you could even use water, although it might freeze as it boils off.