I am considering tacking wire mesh (such as from a window screen) onto the inside of my indoor furnace/AC duct vents as well as the outdoor intake vents, for pest control. I know doing it inside the house might be a little overkill, but I'm wondering if there would be any side effects on furnace/AC performance due to the fans having to work harder to force the air through.
Typical screening can reduce airflow about 50 to 70%. This could destroy the efficiency of your AC and heating units and possibly burn out the fan. Placing any type of screen on intake or exhaust of your furnace could cause it to shut down as it thinks there is a blockage in the venting. Your furnace /AC unit already has a filter on the intake side of the fan so you wouldn't need screening on the individual vents in the rooms.
By "pest control" do you mean insects or rodents? Larger mesh, also called "hardware cloth" would not significantly impede airflow and will stop rodents. Window screening will clog with dust and cause you nothing but problems. If you were thinking window screen to prevent insects, you need no worry about them entering from the outside that way. To get through the ducts and into your house, they would ahve to get through your air filter.