many threads on this topic and NO one consensus approach.
you need to know brand/model and condition of furnace.
the owners packet/install guide on many newer rv furnaces,
will have page on JUST this topic (on/off while rolling)
no manual? it's probably ONLINE.
generally if the suns out and 25 or warmer the unit should be fine...
nite time towing at 30 or colder is different.
((is the plumbing copper or pex or something else?))
((are the tanks metal, fiberglass or polymers?))
((what sort of water pump??))
how tight are the windows and roof vents and misc hatches??
old stuff is different than new stuff (sorta)
here's just 2 threads there are many....
i thing is 4 sure...
if the furnace is used much while towing, the lp gas will vanish quickly.
( i do run the furnace (set to 40) while towing IF the temps are mid/lower 20s)
but trying to keep the trailer 'comfortably warm' while in route is futile.