No, appliances are fine. Flammable liquids vaporize for long periods of time. LP in liquid form vaporizes coming out of the tank or cylinder. It is supplied through outdoor lines up to 10 psi and indoor lines at approx.1/2 psi. Natural Gas is about 1/4 psi. When propane mixes with air it becomes too lean to burn very quickly. So it is much safer than liquids that vaporize slowly.It can become dangerous when the vapor is contained in a building or vehicle and builds up enough to mix with the oxygen to become explosive in a confined area. LP is heavier than air, so it drops to the floor and can be swept out of a building with broom or large piece of cardboard to waft it out. Natural Gas is lighter than air and rises. It takes approx. 70/1 ratio of air and propane for a proper burn, so low in the building is too rich to burn, middle is just right, high in the room is too lean. Since pilots have a safety which closes the safety valve on the gas valve at appliance if the pilot should go out, gas flow stops to pilot and main burner. LP containers are never to be taken into any building. Cylinders installed outside must be 10 feet from any ignition source, so if a wood stove is being used outside, 10 foot is the rule. BBQ grills, electric meter, A/C or heat pump, are all also ignition sources. It is not because the cylinder is hazardous, it is due to filling the tank or cylinder on site when vapor is released into the atmosphere. (The reason for 10 foot from openings into building such as windows below fill valve, doorway, dryer vents, etc.)
One consideration with propane lines inside in close proximity to a wood stove or any heat source is the regulated pressure inside the line increases when heated AFTER regulation. So expanding the fuel vapor in the line increases the working pressure. This is not a problem with constant flow with a pilot operated appliance, but an appliance with no flow when off (called lock up pressure when unit shuts off) such as controlled by an ignition control module allows the lock up pressure to rise and can light harsh with more pressure than normal system pressure when gas valve opens.