Does the computer have a monitor, keyboard, mouse, and web browser installed?
If the answers are yes, then you should keep doing the updates.
If it is a single-use computer, and you and everyone with access to the machine are security conscious (and never make exceptions to "check email" or browse the web, "just this once, for a little bit") then no, you might not have to. You should also run as a limited user, if possible (which mitigates a large majority of the remote code execution vulnerabilities discovered over the past year or two).
Even then, it'd still be a good idea to keep an eye out for bug fixes that you want, though, and particularly egregious security problems.
But, pay attention not to just how you use it, but how others might use it too. Malware doesn't usually come from sites identifiable as "bad" (the so-called dark corners of the Internet). It comes from Yahoo, and the BBC's website, and hacked ad networks spread across hundreds or thousands of sites. It comes from poorly maintained restaurant websites when you're just trying to look up the menu, and so on and so forth.