One hour of TV (depending on the type of TV and the size) could easily deplete the charge on a battery bank. The furnace is "sensitive" to the voltage input. There is a "sail switch" in the air distribution system and if the furnace fan isn't blowing hard enough to push the sail switch to close, the ignition sequence will not happen.
There are other issues that could be occurring, but you didn't say if the furnace operates "normally" when you're connected to shore power. If it does function normally when on shore power, then it's your 12 volt supply that's causing the problem.
Yes, two 12 volt deep cycle batteries, depending on the size and amperage rating, could easily be well below 12 VDC after an hour of TV viewing.
If your RV is still under warranty, you might want to discuss the problem with your dealer. There is a remote chance that the windings or brushes in your fan motor are drawing too much amperage and depleting the battery, but that's really an "outside" shot at trying to blame the furnace. Chances are your batteries just aren't strong enough to do what you're trying to do. Try running all night on shore power, if you can, then it's probably the batteries.
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John
2015 F250 6.7l 4x4
2014 Cougar X Lite 27RKS
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