Is it advisable to put your CV, research statement, teaching statement on your website? I'm a postdoc in computer science applying for tenure-track faculty positions.
I look at the websites of people who are in similar positions to mine  and about half of them include a link to the person's CV, research statement, and teaching statement.
I'm wondering whether I should put these documents on my website.

What are the pros and cons of publicly posting your CV and research/teaching statements on your website?

I see little upside or downside to this. Would this help me on the job market at all? Would there be a potential downside? It feels a little like sharing too much information with the world.
TLDR: It may help slightly, but really doesn't make too much difference.

As with many things in academia, and other fields, different people will have different opinions on this.  I, like you, was never one to want my CV or research/teaching statements out there for the whole world to read, and I never did it, though I had the "most important" information from my CV on my webpage already (e.g., publication list, where I did my PhD).  

From the hiring side, I occasionally look at applicants' webpages (though usually don't) but I look at things like CVs and research statements from our online application system (in math, we use a central site called mathjobs), so it would make no difference for me reviewing your application.  Part of the reason for this is convenience---it's easier to look at everything in one place, and one of the things I most want to look at are your letters of recommendation, for which I have to go through the application system.

That said, I think there can be a slight advantage for putting your CV and your research statement online, and it is this.  If someone is already looking at your webpage (either because they saw you applied, or for random other reasons), and they want to find out more about you, it's easier for them to look at your CV/research statement directly from your webpage, i.e., there's a higher probability they will look at these things.  So there's a greater chance they will get interested in you as a job candidate.  Potentially, this could result in solicitations for you to apply to positions you aren't aware of/didn't apply to, but I don't know how common this is in CS---in math in the US, most types of positions tend to be well advertised and easy to apply to with mathjobs.

But, I don't think this will affect your odds so greatly that you should feel you have to do these things.  (Also, I don't see any disadvantage to posting them, though probably there's no need to post a teaching statement.)  This is especially true if you have the info you want to advertise to most people on your webpage already (e.g., publications, academic history, maybe a short research blurb, ...).