If you're a teacher and you have a Facebook account, which is probably most teachers, you are likely to receive friend requests from your students. Students don't know anything, which is why they need teachers to educate them, and so they may not really understand why this could be a bad idea. As nice as it would be for teachers to be able to just wish these sorts of murky situations away, that won't happen. Sorry. Instead, because this is a issue sensitive to many people, it would probably be best to err on the side of caution and just avoid a Facebook relationship with your students altogether. Easy enough to follow through on that one.
Whether you agree with this path of least resistance and prefer some other course of action so as to amicably resolve the potential problem, there is one thing you should most certainly not do: act shady about being Facebook friends with your students by telling them to keep it on the down-low or, worse, set up fake accounts altogether in order to befriend students.
A couple of teachers in England apparently missed this policy memo and are now being investigated for maintaining inappropriate relationships with students via Facebook. One teacher who, incredibly, exchanged comments with a former pupil about posing for erotic photos over a webcam received a 12-month suspension. Another teacher received a reprimand for using a decoy account in order to interact with students via Facebook.
This isn't exactly breaking news because everybody knows there are creeps on the Internet. That's not even to say that these teachers are explicitly creeps; they could very well be decent humans who just happened to make some very questionable decisions this time. It happens. It's happened in the United States, it's surely happened elsewhere, and it's a pretty safe bet that it will continue happening in places. But if you're doing something that makes you self-conscious enough to try to obfuscate your actions, then what you're doing is more than likely not a good thing.
In the world of journalism, there's this thing called a breakfast test. It goes like this: when determining whether the material you're about to publish is appropriate, you ask yourself, "Would this be too shocking for someone to read while eating breakfast in the morning?" The metric here is that if the material is offensive enough to cause someone to choke on their Cheerios or spit out their bacon, then you probably shouldn't publish it.
Similarly, if you're a teacher, consider how some of your colleagues would pass the breakfast test if they were to discover in the morning news some day that you're being investigated for how you've been corresponding with your students on Facebook. If you think your colleagues might require the Heimlich maneuver upon hearing the news, then you might want to re-evaluate the importance of those Facebook interactions with your students.