What should I do about burnt parts of Qur’anic papers buried under the bedroom tiles, and is it necessary to break the tiles and search for them?

You are not required to break the ceramic to search for parts of burned papers beneath it, because that involves wasting money, and causes embarrassment and hardship. Also, it is almost impossible for them to be present after the ceramic has been laid over them. Searching for them is a futile and pointless endeavor, especially since you are not certain that there are visible Quranic letters on them. Leave things as they are and beware of obsessive doubts (waswas).