If you shut down your sub, power it off or unplug it, then pull the sub cable off, plug the power cord back in and then power on the sub and toggle the power switch on and off and notice if the driver moves at all and notice what sound the sub makes as you you toggle the power switch on and off. With the sub cable pulled and the sub's amp on stand by it should shut off within 10-15 minutes, if not the amp needs replaced. I had a problem with one of my XS30SEs not powering off at stand by for two hours, it ended it being the source which was my Comcast DVR which was the last source used before powering off the receiver, it had a residual noise signal that the sub's amp was detecting through the sub cable and thus the long shut off in stand by. I disconnected the the sub cable and the amp in stand by shut down in 10 minutes.
Your sub's amp not powering down in stand by with sub cable pulled means the amp is defective, the popping when the furnace kicking on is the circuit your sub is on and has nothing to do with the stand by mode not powering off without a sub cable attached. Try a different circuit or receptacle that you know the furnace is not on it and see if the popping still exists when the furnace kicks on, that will at least eliminate one problem. The other problem (sorry to keep repeating myself) with the sub in stand by not powering off without a sub cable plugged in is a defective amp, normal shutdown in stand by is 10-15 minutes. Do you hear the pop when the furnace kicks on with the sub cable pulled and in stand by mode? Also when you shut your AVR off is there a particular source that was used when you shut down the AVR or does it happen no matter what the previous source was used? The furnace situation has me concerned, I wouldn't worry about damaging the sub as I think the damage may already happened. If you get anything out of all that I said the sub amp in auto on/off position should never stay on with the sub cable pulled. Try a different outlet with an extension cord with a similar AWG as the sub's power cord and don't connect a sub interconnect and power it on and then put the toggle switch in the auto on/off position and see what happens, if all is as should be the sub will power down within 15 minutes and listen for the furnace starting and get back with your results. Sorry for the lengthy reply, I have a hard time trying to put my thought into words. I could have told you all of this in 2 or 3 sentences.