The First Amendment does not apply to yelling “fire” in a crowded theatre, or to maintenance of a Website promoting illegal activity such as illicit file-sharing of copyrighted material.
U.S. Immigration and Customs Enforcement (ICE) has a program called "Operation in Our Sites," that seizes domain names connected with Internet piracy.
Seizure of a domain name before trial is called prior restraint. That, like banning publication of a book before trial, is generally not allowed in First Amendment cases absent the establishment at a preliminary hearing of a substantial probability of success in proving illegal conduct at trial.
USDC SDNY ruling:
EFF Outline of Puerto 80 v. US: