While trying to stop the ticking noise coning from my furnace ducts, I accidentally dropped a small piece of fire retardant foam (as you can see in the below photos) in one of the vertical furnace ducts in my basement. I can’t tell how far down it does with my fingers. Is this cause for concern (should I have someone remove it)?
There's a much easier solution
On your furnace you have mounted a Aprilaire MK 500. Inside that humidifier is a 9" x 9" opening. You simply need to remove the cover by by depressing the top or bottom tab, pull the 1/4" diameter water line out of the humidifier pad and then pull the humidifier pad out.
You should now be able to see inside the furnace with ease if the foam has fallen down you can run the fan to blow it up and tried to catch it if it didn't already move downstream. If it's not stuck down at the heat exchangers I wouldn't be overly concerned. If it was stuck down on the primary heat exchanger the worst thing you could expect to happen is it would smell a lot like burning foam beyond that it would be unlikely to sustain a flame but not out of the realm of possibility.
For the record bypass humidifiers are supposed to be installed on the return air but it's not the end of the world and this time it worked out to your benefit.