I was told that if you let the fan continue to run after a furnace shuts off, the fan circulates the residual heat in the heat exchanger, thus making the air warmer to heat the next time and getting the most for your furnace running time.
Life's lesson #386 Stay away from the person feeding this mis-information to you. While furnaces are designed to run for a short period after the flame/burner goes out, It's to cool the heat exchangers down to a safe level. Running the fan for extended periods only costs you money in electrical usage. There is no logic to what you have said. Think about it.......
In the plenum of the forced air furnace, there will be a thermostatically controlled switch that tells the blower when the air that's being pushed out of the ducts has dropped to a temp. that no longer contributes to heating the available air, but rather to cooling it. When it reaches that point it's prudent that the fan shuts down. No idea how you plan on getting the fan to continue running past this point, but it will certainly involve some rewiring and I fail to see where any savings will be had.
there is always residual heat, and if you can capture it, so much the better. You cannot damage anything, so try it. Just like the damper which shuts down the exhaust from the furnace after the furnace shuts off, and co2 is not in play. You can save a lot of money by not letting heat go up the chimney uncaptured. Goood luck
Most furnaces have temp or time controlled fan controls to allow the fan to continue to run after the burners or elements have kicked off. This allows air flow to continue moving around the heat exchanger or elements to aid in cooling them back down. No to your question as to saving money
most furnace have this built in. They run for approx 90 sec. after the flame is out to move the residual heat.