The FBI should be subject to the same regulations as other law-enforcing departments. The ability to encroach on someone's privacy without their knowledge or consent should be made illegal. If the FBI were not subject to these rules they could falsely accuse anyone or someone could be made a victim simply because of false alarms.
The danger to me is if the FBI realizes the difference between terrorists or people who they think are suspicious. They've had a history of casting wide nets with investigation to include innocent people, and adding access to users' browser history, and possibly cache, bookmarks, cookies, etc., then anyone who for any reason than terrorism, like journalist, researchers, students, interested people, ethnic minorities, with any history of visiting suspicious Websites, whether by accident or intention, automatically suspects. I don't trust the FBI with that access with respect to individual privacy and freedom.
At times, the FBI definitely deserves to view internet browsing history. However, this should not be allowed without a warrant. Private citizen have rights. The FBI should not be able to search my computer or my internet history without due cause. This is invasive and should be illegal in the US.
No, I do not think the FBI should be able to access internet browsing histories without a warrant because that is an utter invasion of privacy. I believe that if it pertains to matters of national security, then the FBI should be allowed to access the individual's browsing history, but only in extreme cases.