The furnace should run with NO 110 electric being provided - either generator or shore power...
The blower runs on 12v - the heat source is propane.
Typically air conditioners, microwaves, and TVs are the primary needs for 110AC via shore power OR genny (OR inverter for some)... Some have an option for AC for water heater (but runs on propane too), or heating pads on the waste tanks for cold weather... And of course some have residential refrigerators which need 110 as well.
(In general - if on shore power, there would be no need for the genny running at all... Only exception I've seen was my old 2000 Infinity. It had 30 amp service that could only power one AC - needed the genny AND shore power if I wanted to run both AC units...)
One possibility to not igniting is air in the gas lines... I typically 'purge' the lines after propane has been off by lighting one or more burners on the stove. Typically takes a minute to light - and then the fridge and furnace are happy.
Not yet retired...
Florida (Michigan transplant)
2014 Hurricane 32A
2000 Infinity (previous)