Well, when it comes to the library, there were two lootings of it, one in 1526. after Mohács (at this time, the collection was mostly complete when Suleyman invaded Buda) and one in 1541. when the Ottomans "recaptured" Buda. At that point, most of the works there including approx. 650 antique manuscripts have disappeared. Some of the books were saved by Queen Beatrix who took them to Naples with her, and Mária von Habsburg took some "corvinae" to Brussels. Still, it's important to note that the library was already declining after king Mátyás' death in 1490, as it was mainly supported by his own personal enthusiasm. Mátyás' successors were simply not interested in continuing his work, and many of the illuminated scripts, unfinished, remained in Florence and were taken by random people. Some, like II. Ulászló, claimed that they were "inspired by the desire to improve the library", but this went no further than mere words. Some manuscripts were bought by ambassadors and foreign travelers, and so on.

But back to the topic - it is said that after the disaster at Mohács the Ottomans pretty much walked into Buda without much resistance, and the city was set on fire, but Suleiman spared the Buda Castle, where the library was kept, as he wanted to make his residence there. Nevertheless, this did not prevent the Turkish troops from taking many of its contents to Constantinople, including the treasury of the Hungarian kings, household property, guns and shells stored there... Among the trophies were the most beautiful and richly decorated manuscripts from the Corvinus library. That's how a part of the collection was transferred to Constantinople and was used by Turkish diplomacy for centuries to present their ambassadors with valuable presents, and that's how books ended up in Torun, Oxford, Leipzig, and so on. 26 books were sent to Franz Joseph alone and were kept in Vienna. Additionally, in 1862, members of the delegation of the Hungarian Academy of Sciences discovered in the seraglio the last remnants of a once huge collection - 39 codes. With the permission of Sultan Abdul-Aziz, some of them were presented as a gift to the government of Austria-Hungary and returned to Budapest. The remaining codes were returned by Sultan Abdul Hamid II in 1877

Contemporary accounts also suggest that a part of the books was also destroyed. Oláh Miklós. who spent two weeks in Buda around Queen Mária (in autumn 1527) notes that after the death of king Lajos and the subsequent occupation of Buda some of the books were torn apart, and some had their silver linings torn to be used for other purposes. Brenner Márton, likewise, (in his preface to Lippi Brandolini, written right before the final loss of Buda) describes the events as an example of "Asiatic barbarianism", and notes that the library was destroyed and, when he examined it, there was no trace of its earlier glory apart from one or two Greek authors (paraphrasing from Hungarian). Of course, you have to note that this guy was by no means unbiased, and that there might be some artistic exaggerations here and there, but it's indeed possible that some of the books (including a unique collection of Greek manuscripts) were apparently simply destroyed by the Turks, since they did not appear anywhere later.

Now, I'm not sure if this is still related to your question, but as time passed, there was this growing idea that the library was still in place and many tried to obtain it or at least see it. That's how as early as 1605, the preacher Péter Alvinczi from Kassa went to Buda to see the library of King Matthias, but was not allowed into the palace. About two decades later, Pál Enyedi's song enthusiastically tells us that "even in Buda, Matthias's library is in the same place as it's always been, and that there was no loss but the guards were assigned to it." According to later, not entirely certain data, both Gábor Bethlen and Péter Pázmány made an attempt to acquire the library. On the other hand, we already have quite authentic sources that Prince György I. Rákóczi made four attempts to convince Hussein Pasha and the high Porta to obtain the remains of Corvina for the school in Gyulafehérvár. The attempts of György I. Rákóczi remained just as fruitless as those of the foreign count Althan and Corderius, who asked for the intervention of the pope and palatine Miklós Esterházy to enter the library.

The reason for this growing interest might be the fact that some books remained in the Buda palace even during the occupation, but none of them were particularly fancy. They were first inspected when the Viennese court librarian Peter Lambeck, through the intercession of the imperial ambassador, went to Buda in 1666 with the commission to obtain the remains of the Corvina for the Viennese imperial library. And these books were viewed with great disappointment by the Italian military engineer, Marsigli, among the still scanty ruins of the Buda Palace, immediately after the recapture in 1686. About 300 of those still usable were listed and taken to the Imperial Library in Vienna. They are still there today, many of them have been identified. But it turned out that these books, which survived 150 years of conquest, never belonged to the library of King Matthias, but to the library next to the chapel. They are simple, unadorned, mostly paper codices, mostly printed books, with a few exceptions theological (ecclesiastical law, holy speeches, liturgical books). This is probably the exact reason why they survived, they were so modest that they weren't worthy of being brought to Constantinople in the first place. Needless to say, after these events the interest in the library diminished and the remnants of the library were scattered all over Europe. Nowadays there's a virtual reconstruction of the library going on.