PC to TV connection goes black when furnace or stove turns on
You said as a second monitor. 

This typically happens when there is no signal to drive the monitor. Your PC is losing power via a "brown out" caused by a greater current draw. 
Is the TV plugged into the same outlet as the PC or a different outlet ?
Your HDMI connection -is it a cable or a HDMI transmitter , if it is a cable - How Long is the Cable ?

Measure the voltage in your PC's outlet, do the same for the TV (maybe you have a splitter - 6 way or something) make sure you measure with the components ON - that voltage should read above 108V [USA], if it is not your line voltage is low (check other outlets to be certain), the cause could be something in the home or something with the incoming power - the power company will need to boost your voltage via a tap if the latter is the case. 

The reason the TV sees the problem and not the monitor - I am guessing you have a cable, the cable is probably 12 feet long, lots of resistance - lower signal level (voltage drop because of resistance). Furnace kicks on draws a heck of a lot of current (blow dryer on high might do the same), then your TV goes dark because the voltage dropped in the PC which was driving the HDMI across that cable, or your TV input for the HDMI dropped level or both, hence black flickering. 

Get a 3 foot cable or an HDMI transmitter/receiver, that might tell you all you need to know - given the above explanation.

EDIT 3/7/2017

It is a voltage drop caused by the current requested by the furnace or oven. So that is the real culprit. It hi-lites the secondary issue you are facing one of your devices is unable to sustain proper regulation of power during the drop - I suspect the PC (a new PC power supply might be in order) - but it could as well be the TV, and it could also be that both have difficulty. Testing the AC line voltage level is the first thing I would do - if the voltage is 110 or lower - you might want to call an electrician or the power company and have that fixed. While 110 is acceptable - the furnace / oven kicking in drops that - and that reason alone it needs to be fixed (how is that done - the power company taps your transformer up a notch.)
In reality this dropping is bad for all of your electrical devices. 
Plug an INCANDESCENT lamp into the computer outlet or tv outlet does the light dim when the furnace or oven comes on ? [don't look at the light directly -just look to where you can notice if it gets dimmer for a split second, like a quick flicker. - Make sure it is an OLD SCHOOL INCANDESCENT not an LED or Compact Fluorescent].