Are you allow new air into the balloon (as in a regular hot air balloon) or are you assuming a sealed system as you have drawn?
A regular hot air balloon where you heat outside air will run as long as you have fuel.
A sealed system will require you to constantly increase the temperature to maintain a constant pressure and so a constant volume as the amount of gas present decreases due to leakage. Ultimately it will be limited by the temperature the envelope can survive or the breakdown of the gas molecules.
I suppose in theory if you had a envelope that was completely heat proof and you were able to heat the gas inside to a plasma you could hold the balloon up for a long time!