Wayback Machine will remove archived tweets. For example, Taylor Lorenz’s tweets were erased after she requested it.[1] Other archives get removed if the people running the Internet Archive dislike the content. Similar to how Paypal isn’t a bank and eBay isn’t an auction site, The Internet Archive isn’t an archive.
Tweets are subject to copyright so I doubt the Wayback machine has any legal leg to stand on for continuing to host someone's tweets after a takedown request.
When copyright expires in 125 years or whatever if the archive still exists they can publish them again.
1. https://web.archive.org/web/20220000000000*/https://twitter....