Pogre
I am recording with android cell phone and playing with speaker. When I pay rain sounds on my cell phone its too loud for me to fall asleep. That is why I like furnace noise.
Using cell phone microphones and speakers is generally going to get you a nasty sound with exaggerated tinny highs!
I like the HEPA filter plan and also, have you tried running the fan from your heating unit. For most central air systems, you have a choice of FAN Auto (turns on/off with the furnace) or Fan On (stays on continuously). There are lots of variables that determine how much of what you hear from your Furnace is the fan Vs the sound of the gas flames, but it is simple enough to turn the fan on and see how well that works.
If it is your place and they are the tenant (ie - you have control over their space), you could also add some carpeting or area rugs (assuming it is currently a bare floor) to help deaden the sound from their space.