I think I might be able to "prove" otherwise through the use of LOGIC.
If the girl doesn't like you in that way, then regardless of whether or not you let her know you like her, you are likely to be in the friend zone. I say likely instead of definitely because she might magically like you if you ask her out, (which I think is kind of shallow if you not asking her out is enough to make her ignore your personality traits, but whatever). If she does like you in that way, and you let her know you like her, then you are not likely to be in the friend zone (no shit, right?). And, if she likes you and you don't let her know, then one of two things will probably happen: She makes her move on you (because she likes you), or she doesn't and she gets friend zoned. And then you are both in a friend zone equilibrium.
Now, don't misunderstand the point of my "proof." You asserted that if you don't let her know you'll be in the friend zone. What I'm saying is that isn't the only option, and what's more isn't the only logical option when all other scenarios are considered. If she does like you back, then it IS possible for you to get out of the friend zone, regardless of whether or not you tell her you like her. You might go with "girls don't want to ask guys out so it's more likely that you'll be in the friend zone." While there's probably something to that because of biology or whatever, I don't think this is necessarily true and don't think the biological context kicks in for this situation- in a situation where you two are friends and close with each other and she already likes you romantically.
I think that's more true when the friendship isn't established, and a girl and guy are meeting for the first time. It's then that it's unlikely for a girl to ask a guy out. But logically, if they are friends and are repeatedly exposed to each other, and the girl likes him romantically, the chances that she'll eventually break get higher and higher- or at least start becoming on equal ground with that of the guy, like with how the probability of life in the universe, despite being small, had a decent chance at happening due to the sheer amount of times (or high probability for) the potential chance to even happen, through the huge amount of planets or whatever. So with that in mind, the idea that the guy has to be one to move out of the friend zone isn't exactly true. Though I'm in no way defending those who are too much of a pussy to let the girl know their true feelings. In fact, the only meaningful definition of the friend zone in my eyes is when the two have already done so.
Also, regardless of how one feels about the friend zone, I think this video is pretty interesting to watch:
True, there is the factor of the girl liking the guy back that I didn't consider. From what I've seen and learned through others however, it is very
unlikely that a girl will ask a guy out first. Really, it depends on the person, but nowadays it's just the norm for males to make the first move. Though there are exceptions to every rule, of course...
That idea of "friendzone equilibrium" is actually quite tragic... Granted, you have that simple benefit of having a great friendship with someone you admire, but when mutual romantic feelings are involved, it seems almost cruel that two people would be prevented from being with one another due to their own inability to let the other know how they feel. In that scenario, if one person confesses to the other, it wouldn't be unreasonable to think that the other would admit to their own feelings.
The girl-likes-boy scenario seems like the only factor that would actually discount the idea that if you don't let the girl know how you feel one way or another, you will be friend zoned. Even a subtle hint can let a person know that you have an interest in them, and for most, that small spark is enough to spur some sort of feeling towards that person. It's natural to like someone whom you know likes you back more than someone who doesn't (unless there's an intense infatuation involved, rather than sincere feelings), though as I said, it's largely dependent on the person. Everyone is different and responds to things differently. ^_^
I'm going to watch that video right now.
Edit: That was fantastic. I had no idea there was a "scientific" explanation. I love how he veered off the main "friend zone" topic and discussed the way that use of the internet and social networking causes a decrease in people hanging out in the real world.