If it's a modern LED TV you are good to go. I do exactly what you are asking. But, as stated if you run it for too long you will drain your battery. I put 2-deep cycle golf cart batteries on my TT. I converted all the lights to LED. This saves a lot of electricity right there. You can put a switch into the constant hot line for your radio to save some power. (I didn't need to because I have a old analog radio with no memory circuit.) Then by only running the TV about a hour or so a day I can go a easy 4 days without plugging in. Maybe longer I've never been out longer then that so I don't know. You refrigerator will always use some power even on the gas setting that can't be helped. And, most importantly of all you really can't run the furnace at all. The furnace fan will drain your battery/s quite quick.
