According to Quran and Hadith,
If a Muslim desecrates Quran, what punishment is prescribed?
Also, what does this heinous act say about his Muslimhood?
If a Muslim desecrates the Quran then he commits kufr and becomes an apostate. This should be self-evident. Because desecration of a text implies that the perpetrator disagrees and condemns its contents, which is kufr by definition.
Further, for example, mocking and hating the Quran has been explicitly declared as Kufr. Desecrating a copy of the Quran is a way of expressing mockery and hatred for the revelation and verses of Allah.
قل أبالله وآياته ورسوله كنتم تستهزئون لا تعتذروا قد كفرتم بعد إيمانكم
Say, "Is it Allah and His verses and His Messenger that you were mocking?" Make no excuse; you have disbelieved after your belief.
والذين كفروا فتعسا لهم وأضل أعمالهم ذلك بأنهم كرهوا ما أنزل الله فأحبط أعمالهم
But those who disbelieve - for them is misery, and He will waste their deeds. That is because they disliked what Allah revealed, so He rendered worthless their deeds.
إذا أهان المسلم مصحفا متعمدا مختارا يكون مرتدا ويقام عليه حد الردة وقد اتفق الفقهاء على ذلك
If any Muslim intentionally and willingly disrespects a mushaf then he shall become an apostate and the hadd punishment of apostasy will apply to him. This is agreed upon by the jurists.
The punishment for apostasy is execution:
من بدل دينه فاقتلوه
Whoever changes his religion, kill him.
— Bukhari
Next, as far as desecration done by non-muslims is concerned:
Islam voids pacts (such as peace treaties and dhimma status) and prescribes fighting against those who revile the religion:
وإن نكثوا أيمانهم من بعد عهدهم وطعنوا في دينكم فقاتلوا أئمة الكفر
And if they break their oaths after their treaty and defame your religion, then fight the leaders of disbelief
( وطعنوا في دينكم )
أي : عابوه وانتقصوه . ومن هاهنا أخذ قتل من سب الرسول ، صلوات الله وسلامه عليه ، أو من طعن في دين الإسلام أو ذكره بتنقص
(and attack your religion...) with disapproval and criticism, it is because of this that one who curses the Messenger, peace be upon him, or attacks the religion of Islam by way of criticism and disapproval, they are to be fought.
Desecrating the Quran is an extreme form of reviling Islam, hence it would fall under the same ruling.
Islam also prescribes taking precautions to prevent desecration of the Quran. Among which is that selling a copy of the Quran to a non-Muslim, or carrying it to a hostile non-muslim land is forbidden. This being a precaution against any non-muslims disrespecting it.
أن رسول الله صلى الله عليه وسلم نهى أن يسافر بالقرآن إلى أرض العدو
Allah's Messenger (ﷺ) forbade the people to travel to a hostile country carrying (copies of) the Qur'an.
— Bukhari