[Raed667]

At least, for domains that you own, yes they do. I have asked them once to remove some links from the archive because of accidental private data being exposed. The process was fast and straightforward:

1- Add a page or DNS record proving that you own the domain

2- Email receipts showing that you owned the domain for the requested range

3- Wait 24 hours

[drakonka]

They also comply with this for social media accounts on domains you don't own if you can show somehow that it's your account. In my case, it was enough to email archive.org with a Twitter profile exclusion request with the same email I publicly listed in said Twitter profile.

For my domain, it was also enough to email them from the email address listed on my site's contact page without any DNS changes.