You can learn to play the violin, or near enough any instrument, with enough time, patience and practice - teacher or no teacher. But in my experience, the thing that really falls short in many instruments, but especially the violin, without a teacher is proper technique. Without decent tuition, you'll in all likelihood end up holding the bow wrong, the violin wrong, end up using your hand rather than your chin to support the weight, and so on.
The main big problem with this isn't that it causes major issues straight off - because it doesn't. At least, not noticeably major issues. This is in fact the big problem, because you can go on with a crappy technique for years before you start to notice it becoming an issue, but when it hits it's going to really hit, and it's going to be incredibly difficult to un-teach yourself and gain the right technique, even with a proper teacher. Sure, it's possible, but it's damn hard.
For example, without a balanced bow hold, it's nigh on impossible to get a good, fluid staccato, and could really cause your arm to ache after a while of hard playing. You'll struggle to get a smooth, constant sound as well. A bad violin hold will make shifting positions much harder (again, something you don't do when you first start) and make tuning much more inaccurate. You'll also struggle to move your fingers quickly and accurately if you're supporting the weight of the violin too.
So while you don't need a teacher if you want to play the violin, I'd say it's definitely worth it - especially if you're planning on reaching a decent level.