I know after Napoleon and his army won the battle of Austerlitz the Holy Roman Empire and Emperor officially were dissolved and Francis II became simply Emperor of Austria. I also know that today the crown sits in the Imperial Treasury in Vienna. But how did it get there? Did Francis just take off the crown and put on another one or did he continue to wear the crown as the Austrian Emperor? Why didn't Napoleon just take the crown for himself as a trophy?
The Imperial regalia, including the Reichskrone, used to be kept in Nuremberg and only taken out of the city for coronations. During the Rhine Campaign of the Revolutionary Wars it was feared that the French might take the city so it was decided to move everything out of the city, first to Regensburg, but as that was also in danger it was decided to ship it all off to Vienna for temporary safe keeping. After Austerlitz Francis II abdicated and also declared the Empire to be dissolved, he had raised Austria as an Empire two years previously in response to Napoleon declaring himself Emperor. For his newly created Austrian Empire he used another crown that was made in the 17th century (which was used by Emperors as their sort of personal crown) as the Imperial Crown of Austria, so in 1806 just before dissolving the HRE he already had two Imperial crowns, one for the HRE and one for Austria.
The crown actually moved again later. After the Anschluss, the annexation of Austria into Germany by the Nazis, Hitler decided that the crown jewels should return to Nuremberg. They were displayed in the town hall until fears that bombing raids might damage them led to them being moved to an underground bunker along with some other artworks. The crown jewels were returned to Vienna in 1946.