I thought that you earn rent regardless of whether or not your console is on...no. however you need not leave your profile logged in. you can collect rent even while your box is turned off. you may also change the system time of your xbox console to gain rent for that time. there is a maximum gold per collection of approx 100x the rent you would normally get every 5 minutes.
I believe you gain rent even if the game is off as long as you are connected to xbox live. If you choose to leave your console over night then the game will stop as soon as your controller shuts off and you will stop gaining rent.
you dont stop earning rent even if the games paused as it accumulates, if your going to leave it on then i suggest pausing it.
the easiest way to gain rent is to sign out of xbox live, save your game, then go to the xbox dashboard and put the time and date forward. then reload your Fable II game. this is a good way of getting a lot of money quickly, and you don't have to leave it alone for ages.
EDIT:tried to restore it to its previous state but it may not be exact NOTE:vandalism will not be tolerated and can result in an i.p banning