When and how did Arjuna's Chariot bursts into flames?
Who had burnt out Arjuna's Chariot?
What happened to Hanuman who was sitting on the flag of this Chariot?
At the end of the Mahabharata war, Krishna told Arjuna to get down from the chariot first, then Krishna got down, and the chariot burned to ashes. That's because the chariot had been struck by so many divine weapons over the course of the war, but Krishna's presence was keeping the chariot together. So as soon as Krishna got down, the deferred effect of those Astras finally took hold, as described in this chapter of the Shalya Parva of the Mahabharata:
Arrived at the pavilion of the Kuru king, the Pandavas, those foremost of car-warriors, O monarch, dismounted from their cars. At that time, always engaged, O bull of Bharata's race, in the good of his friend, Keshava, addressed the wielder of gandiva, saying, "Take down thy gandiva as also the two inexhaustible quivers. I shall dismount after thee, O best of the Bharatas! Get thee down, for this is for thy good, O sinless one!" Pandu's brave son Dhananjaya did as he was directed. The intelligent Krishna, abandoning the reins of the steeds, then dismounted from the car of Dhananjaya. After the high-souled Lord of all creatures had dismounted from that car, the celestial Ape that topped the mantle of Arjuna's vehicle, disappeared there and then. The top of the vehicle, which had before been burnt by Drona and Karna with their celestial weapons, quickly blazed forth to ashes, O king, without any visible fire having been in sight. Indeed, the car of Dhananjaya, with its quick pairs of steeds, yoke, and shaft, fell down, reduced to ashes. Beholding the vehicle thus reduced to ashes, O lord, the sons of Pandu became filled with wonder, and Arjuna, O king, having saluted Krishna and bowed unto him, said these words, with joined hands and in an affectionate voice, 'O Govinda, O divine one, for what reason hath this car been consumed by fire? What is this highly wonderful incident that has happened before our eyes! O thou of mighty arms, if thou thinkest that I can listen to it without harm, then tell me everything.' Vasudeva said, 'That car, O Arjuna, had before been consumed by diverse kinds of weapons. It was because I had sat upon it during battle that it did not fall into pieces, O scorcher of foes! Previously consumed by the energy of brahmastra, it has been reduced to ashes upon my abandoning it after attainment by thee of thy objects!'
As far as the monkey flag goes, as the passage describes, after Krishna got down, the flag disappeared, and only after that did the chariot burn to ashes. So the flag may have also been protecting the chariot from burning to ashes.