Now I urge you, brethren, keep your eye on those who cause dissensions and occasions of stumbling contrary to the teaching which you learned, and turn away from them.
There are some who think that becoming a believer means that we become naïve and trust anyone and everyone who comes along claiming to represent God. That is just not true. Jesus commanded us to examine teachers and people who claim to represent Him and gave the litmus: by their fruits we will know them.
Paul reiterates Jesus’ instruction in another way. This is best practice in teaching. Teach a concept, then teach it again another way, and repeat until the concept is grasped. Paul tells the believers in Rome to keep [their] eye on those who cause dissensions and occasions of stumbling contrary to the teaching [they had] learned. This implies a few things: (1) that the Roman believers knew who the troublemakers were; (2) that the Roman believers could locate those they did not already know about by their wake — i.e. dissensions and occasions of stumbling; (3) that the Roman believers had a good foundation in sound doctrine. All of these are necessary components to recognizing those who are just in the fellowship to cause problems.
Paul does not stop with the believers merely keeping an eye on those individuals, he instructs the believers to turn away from them. Often, controversial people are controversial merely for the sake of getting attention and drawing a crowd. It is not always the case and would be too facile an answer to say that it is usually the case, but it is often the case. Often enough that [turning] away from them is a viable method of addressing their wrongdoing. Take away their attention and their potential crowd and they will either stop or move on to some other place where they can rouse the rabble.
As a believer, I am called to be judicious in who and what I allow to rile me. If something is genuinely offensive to God, it can rile me. If someone is misrepresenting God or making it difficult for others to come and commune with God, then that can definitely rile me — it riled Jesus enough for some tables to be flipped in the temple. If what a person is saying and doing is causing dissent (disagreements, usually angry and often baseless) or giving other believers reason to stumble, then I am to turn away from them; have nothing to do with them. I am not to call them out or repudiate them or try to defend God or stand up for the truth as so many often claim, but rather am to have nothing to do with them. This was illustrated marvelously a few years back when I was asked by a fellow believer what I thought of a purportedly Christian writer. I had never heard of him. I read a bit of his work, put it down, told my fellow believer that the man was leading people astray, and then went on about my life. I would be hard-pressed to remember who he is, now. That, I think, is the idea. Examine what they have to say, if it is merely contentious or a cause for stumbling, then have nothing more to do with them. If there is merit or some digging into the Word and seeking answer through prayer is called for, then do so and act according to what God reveals.