Unless your problem is a bad/sticking thermostat, then no, removing it won't help. And if the t-stat IS the problem, then you might as well just put a new one in while you have it "apart" to remove the old one.
Running with no t-stat will lower your MPG, as well as increase your emissions, which could be a big concern for you, being in CA. It will also increase wear and tear on your engine. Further, in cool weather, it will take longer for you to have heat. All of this is because the thermostat is intended to restrict coolant flow when the engine is cold, allowing it to reach operating temp quicker. Also, it helps to keep it at operating temps as you go down the road. Running w/o a t-stat causes the engine to take longer to warm up , and can prevent it from ever reaching proper operating temps, even in a warm location like CA.