Tim: disconnecting the audio input interconnects from the amps tells you the amps themselves are picking up the "clicks". Could be incoming via the AC power, or could even be airborne RFI from the furnace highvoltage arc. You already switched to the opposite AC phase which is a good thing to try. A high current AC line isolation transformer may or may not help, but you could try that if you're committed to keeping the tube equipment.
Ensure that the furnace is well grounded. You might try running a temporary ground cable draped across the floor, directly from your distribution panel to the furnace chassis. You could also try connecting a 30uF 400VAC filter capacitor across the AC Hot to Neutral at the furnace (available at your local electrical supply house) which won't cost much and is easy to do. Ferrite filter rings (such as Audioquest sells) placed around the AC line conductors at the furnace and/or at the panel on the dedicated power circuit feeding your amps is also inexpensive & easy to try.
I am not surprised that tube equipment would do this because tubes have extremely high input impedance, making them highly noise susceptible. The BelCanto's being solid state are current mode devices which are far more noise immune. Your only fix may be to eliminate the tubes, which is certainly what I would do.
You make no mention of AC line conditioning which might be a viable cure; borrow a line conditioner to try it. Some listeners don't care for line conditioners on amps, but there are high current models available that won't adversely affect your sonics. I use line conditioning on my whole rig, not only for noise supression but also for AC line transient protection. Catastrophic experiences with lightning damage have previously resulted from my not using any protection in the past; repairs are quite costly not to mention long downtime.
I have a "Chang 9900 Amp" high current 20 amp AC line conditioner sitting here unused if you want to try that let me know via email.
