I recently had a new propane water heater installed. At the time of the installation, the plumber told me that the recirculating pump was not working and that this would cause my gas bill to be high. I still have hot water immediately at every faucet in the house, so I question if the pump is bad. It is a Grundfos pump and the installation was done when the house was built (1991), so it is the type with the extra return line, not the type that can be added to any water system. Can you tell me why a failed pump would cause in increase in gas use? Would I still have immediate hot water, if the pump was not working?

Recirculating pump won't affect gas bill to any measurable degree. With out pump you have to purge the line of cold water each time you want hot water. Probably cause a slight increase in water usage. Because water pipes are not usually insulated to the degree that the water heater is, there is a slight loss of heat in the water in the line after you use it. With the recirculating pump there is a slight loss of heat in the line all the time, while it sits and waits for you to use it. Therefore, having the pump will increase your gas bill if anything. If you used the faucet only once a day it would be more economical (gas wise) to not have the pump.
As to why you have hot water even though you pump is not running. There are other things going on. One is the natural hot water rises and cold water fall falls. The water in the tank is hotter than the water in the line. Therefore the hot water in the tank will tend to rise up into the line and the colder water in the line will tend to fall back into the tank. This is a very slow circulation and will eventually stop because all of the water in the line will be reach the same temperature.

There is another phenomena that is occurs called the Bernoulli principle. The Bernoulli principle says that as the velocity of a fluid increases the pressure decreases. It's the thing that makes airplanes fly and carburetors work. As water going into the water heater passes the tee that connects the recirculation line, it creates a low pressure in the recirculation line and the circulation of hot water starts again.

When I built my home I put in a pump and recirculation line to the second floor bath. Water heater is in basement. After a couple of years the pump quit. I remove it in anticipation of replacing it. For once my procrastination paid off. I noticed that I still had hot water at the upstairs bath. I never replaced the pump. I have noticed that once in a while, first thing in the morning, there is no hot water at the upstairs bath. The circulation has stopped. As soon as hot water is used anywhere in the house, (Bernoulli principle at work) the recirculation begins again and I have instant hot water at the bath.

I have the best of both worlds.