Well it really varies on a case-by-case basis. Often enough the institution isn't being forced to change but rather decides to change as it evolves to better serve the public.
The other big issue is that the most concerning side of book burning is in its prevention of you and me from being able to keep content deemed problematic, while those websites are more about what institutions are holding. It's the difference between me stopping you from having what you have versus using the public to give you politically acceptable messaging.
Institutional speech and private speech have some significant differences.