My understanding is that if it is your domain, and you add a robots.txt, then you can exclude it from the web archive.
And this unofficial page suggests that if you forgot to add a robots.txt, but manage to retroactively prove that you owned the domain at a given date, then they might also remove it: https://www.joshualowcock.com/tips-tricks/how-to-delete-your-site-from-the-internet-archive-wayback-machine-archive-org/ and this is also suggested at https://help.archive.org/hc/en-us/articles/360004651732-Using-The-Wayback-Machine:
How can I exclude or remove my site's pages from the Wayback Machine?
You can send an email request for us to review to info@archive.org with the URL (web address) in the text of your message.
But what if you have a personal profile on a social media website such as Twitter, Facebook, Stack Exchange, etc. which has a robots.txt that allows archiving?
Also interested in the special subcase where that content has an open license such as CC-By-SA.
Do they remove archives of your profile upon request?
Obviously supposing that the content is legal according to the laws of where Web Archive is hosted.
Best Answer
I have personally witnessed a case where such request was made with rationale that it would be a personal security risk to the owner of a social media profile, and the specific profile page was "excluded from the Wayback Machine" when I later tested it, and I am certain it had been archived earlier. I cannot provide more information however due to privacy concerns of that case, but know that in that case at least, there is precedent.