Did he burn it down or seek any revenge on the school?
The Vienna Academy of Fine Arts continued to exist from 1938 to 1945 and still exists today.
After the Nazi take over of power in Austria in 1938 there were, of course, changes implemented in the school, the most notable one, a complete purge of all personnel and students that were Jewish or known to be politically anti-Nazi. Immediately after the so-called Anschluss in March 1938, the Vienna NSDAP put three new men in charge of the Academy, Ferdinand Andri, Wilhelm Dachauer and Alexander Popp. Interestingly enough all three of them were aesthetically and politically close to the Vienna Secessionist group and had in the past had personal ties to Klimt.
They immediately began a purge and in the first days after March 12 already 13 Jewish professors were let go from their posts (almost everybody working at the Academy was a state employee since the Academy was and still is a public university run by state employees). Furthermore, the trio (of which Popp later became chancellor of the school in 1941) prepared a manifesto about what was in store at the academy. The new program consisted mainly of pushing a more "German" style of art, cutting ties to previous more abstract art styles within the academy and place a larger emphasis on the monumental. It basically lead to what we today recognize as typical Nazi style of art being thought.
What is also interesting and has only been revealed by recent studies is that the Academy "lost" only 20% of its personnel and only 7% of its student to the Nazi purges. This means that by the time the Nazis took over in Austria, Nazism was already a rather present force within the academy, especially its students. In light of all this it is hardly surprising that the Academy retained its prestigious status and continue to move forward, albeit with a slight change in the art style that was taught as prominent.
In looking through the literature, I could not find any reference to Hitler's thoughts about the academy once it together with all of Austria had become part of the Third Reich.
Sources:
Verena Pawlowsky: Die Akademie der bildenden Künste Wien im Nationalsozialismus. Lehrende, Studierende und Verwaltungspersonal. Vienna 2015.
Oliver Rathkolb: Nationalsozialistische "Kultur"politik in Wien 1938-1945.
A follow up question: is there any actual evidence that Hitler's art school rejection contributed to his violent ideology in any way? There's a popular image of Hitler turning to Nazism because he was so angry about being rejected by art school. Is there any evidence that's true?