This is definitely a site hack. The error message is because once you've started sending the webpage contents to the user, you can't go back and set cookies and such. In vBulletin, various things are processed, including cookies, before any of the webpage is sent. But if someone hacks one of the vBulletin files to output something to the user before all of those things are taken care of, PHP will throw a minor fit and produce that error message.
The offending script probably only does something some of the time (possibly randomly, possibly based on your referrer, possibly based on whether you're logged in, etc.), which is why it's difficult to reproduce.
But anyway, for the administrators:
The first thing to do is check your site for the uploaded third-party scripts that are mentioned in this error message. If you get the error message and view source, you'll see at the very very top that an extra <script> tag appears there, referring to a script at "/uploads/13335/swfobject.js". Delete that script, first and foremost. (You could also replace it with an empty file and set the permissions to read only.)
Next, that line is being inserted because one or more vBulletin files was modified at some point. Which one it is can be difficult to tell, but I would check "/includes/init.php" first, since the error message mentions it. If you don't find anything there, and you have access to file modified dates, then the file in question will have a very recent modification date.
Most likely, the change to the file will be at the very top or bottom. It may just insert that exact line of HTML verbatim.
There may be changes to multiple files. It may be worth copying over the entire vBulletin installation with a fresh one to fix all of these changes en masse. If you've installed a lot of plugins, these should be reinstalled as well. Backup your MySQL database first. Don't forget to delete the /install directory when you're done.
For safety's sake, I would recommending setting permissions to read-only on the / and /includes directories and their contents, since those are where these hacks tend to land. I personally don't know whether these hacks are the result of a vBulletin vulnerability, or if there's some hack on your shared hosting provider caused by a vulnerability in someone else's site (or the provider generally) that gives the attacker access to all the shared sites on the server. (I've had similar problems on forums I've managed and the best I could do was set various directory/file permissions to read-only.)
Oh, BTW, eventually Google may figure out that your site is hacked and add a "potentially dangerous" note to all of your search results. Don't let that happen - get this problem nailed down ASAP.
Best of luck!