Yes and no.

The Archive is very responsive to content owners requesting that their property is removed from the site (either by sending a request to The Archive or by modifying their own robots.txt; the Wayback Machine checks the live robots.txt before serving up old content and uses it as a filter).

On the other hand, there's no law forbidding companies or individuals from keeping a local-only copy of others' IP for non-public use, so IA doesn't actually delete content. They "darken" it instead, which means it's still in the data cluster back-end but inaccessible from the outside.