DMCA Copyright Infringement
The Digital Millennium Copyright Act provides one pathway for removal of documents. These complaints request removal of content from the search results due to rights violations and protected ground, not because of disagreements with the content, message or relevancy. Search engines will not alter search results without a legal mandate.
Links Below Are To Chilling Effects Documents
If you want to understand how to get documents removed from the search, these links to stories on recent DMCA complaints can give you a sense of what others are doing.
It is extremely difficult to force search engines to remove content from the search because they are protected from liability by the First Amendment. Ranks are an opinion, and therefore protected speech. This defense is now a recognized legal precedent. But if the search results of any search engine infringe, or aid in the infringement of rights, a DMCA complaint will usually get the documents removed from that search engine.
These are the most recent DMCA (Digital Millennium Copyright Act) complaints filed with search engines. Note how dominant Google is in receiving these complaints.