Why doesn't a hot air balloon burst even if we keep heating the air in balloon for a long time , shouldn't the air pressure become so high to resist? I don't get one thing: we say a hot air balloon rises as hot air is less dense than  cooler air.
However, if the heat source continues to add heat to the air in the balloon, doesn't the hot air become so hot that pressure will be too high for the balloon to resist and burst? But it doesn't burst even if we continue to heat the air in balloon even for a long time?
 A: 
doesn't the hot air become so hot that pressure will be too high for the balloon to resist and burst?

If you have a sealed balloon, and you heat it enough, then yes.
Hot air balloons aren't sealed though. If the pressure gets too great, they can vent out of the opening where the flame goes.
Even with that though, you'd need to heat the air to the point where the balloon bursts.  There's a lot of air in a balloon, and a large surface area.  However much heat you put in, there's going to be an equilibrium point where you can't heat the balloon any more because all the heat you're putting in is dissipated into the surrounding (cooler) air.  You can (in principle) do the sums to find where that equilibrium point is, but the main thing to note is that if that equilibrium point is less than the pressure which would explode the balloon, then the balloon stays intact.
A: The ideal gas law states that $PV=nRT$. In many cases you apply this to a closed system with a fixed quantity and volume of gas, so you often see that as temperature ($T$) increases, so does pressure ($P$). A hot air balloon is not sealed or pressurized, however - as the temperature increases, air escapes the balloon. If you consider just the air in the balloon, as $T$ increases, the number of air molecules in the balloon ($n$) decreases, resulting in a system of constant pressure and volume.
Air is forced out by a slight and transient pressure differential. Anytime the pressure in the balloon is higher than outside, air will flow through an opening in the balloon to equalize the pressure. The bigger the pressure differential, the faster air will flow to equalize the pressure, so the internal pressure is always close to (but perhaps slightly higher than) the external pressure, and in an equilibrium state, they are identical.
A: Since the balloon isn't sealed and the bottom has an opening, air will escape through this opening thus preventing the point where the pressure inside the balloon is so high that it would explode.
A: Hot air balloons take advantage of the density of hot air, like you correctly mention in your post. Pressure won't cause the balloon to suddenly burst given that the hot air balloon has an exit present at the bottom, resulting in what could be approximately equal pressure inside and out, ruling out a possible fail in the system.
