The DCMA provides exceptions for caching. But in order to be a cache you need to eventually and automatically reflect the wishes of the source. So, if the original content disappears it has to disappear from the cache too. Or, if the original content changes it has to be reflected in the cache too.

In the case of twitter, there is no equivalent for a per-account "robots.txt" and issues like this arise where you can't take your content back once it's copied. Twitter thread compilers should regularly check whether a tweet has been removed etc to turn into a cache and avoid these situations.