. . . at least none that I’ve experienced do.
Why should they? www is a prefix designating “world wide web” service. Not that that’s all it provides (my own has scp, ftp, pop3 and more services), but it’s traditional.
For the SDMB, it allows the straightdope.com domain to have separate machines for the straightdope (www), and for the boards(boards). Don’t tell me you don’t think it needs it, either. 
The convention of naming of web sites with “www” at the start is just that, convention. It started up when the internet was much more of a mix of different types of access and web sites were new. The “www” helped people tell how they should connect to a site.
But there is no reason why they can’t start with anything you like, as long as the end bit is the domain name that everyone else recognises as being owned by you. So boards.straightdope.com could just as easily have been called tomato.straightdope.com, if that’s what the Straight Dope site managers wished.
I suppose that most message boards are not titled ‘www’ to differentiate them from other types of websites.
It’s convention and practice only. They can start with “www” or anything else, really. And of course they don’t have to have a URL at all, just an IP address - hover your mouse over the “www” button below my post here and see one address of my Board.
And there are some that do start with ‘www.’
Plenty of message boards start with www. Here’s a couple off the top of my head: Straphangers and TiVo Community.
The www is just the name of the server that the domain sits in. The www is known as the third level, and it can be whatever the site admin wishes to call it. Since it is the most common convention, that is what most sites use.