I think this comes down to some change (or lack thereof) that's been made on your end.
I could tell you exactly what the error means but I don't think it'll help all that much. So...
Here's one source of information, but let me explain it in my own words: Basically headers are really picky. There cannot be any html, even whitespace, rendering before it's called. If it's rendering, say, an error message because it's looking for improper or incomplete config data, well, you'd get a header error instead of the error that would tell you that's what's wrong, unfortunately.
First off, make sure your
inc/config_forums.php has the correct information AND that it should also go without saying that you also
should have installed the forum by visiting it's url and creating an admin account for it - its not as simple as just dropping the files in. The prefix being correct is the most important part. The file I talk about below is going to look inside your database for data from a forum install with that prefix. So if the config data is correct and the forum properly installed, you should be able to log into the forum by visiting its url with the admin account you just created for it during it's install process. If all this is good, then it's not the problem.
Inside functions/functions_forums.php, which relies upon the information above being correct, the line the error is happening in is the function mybbsetcookie() which is called in mybbregister(), also in that file, from the file register.php. This should be working just fine - so long as nothing is being rendered to the page BEFORE this happens. Can you verify for me that absolutely no changes have been made to
register.php by putting in a copy from a fresh version of the files?
But this isn't the only place where problems could have occurred. Afterall, $mysidia->user->register(); is called before mybbregister() and if it attempted to render anything to the page, even a blank space, this error would be happening. I'm pretty sure that function is being called up from it's location in classes/class_visitor.php (since the visitor class is what actually registers users and extends user) and it's register() function. All this function should be doing is adding a user to the database. Definitely not writing anything to the page, so it shouldn't be interfering. Put in a fresh copy of
classes/class_visitor.php to confirm no changes were made here from the default?
If you've literally made no changes at all (or if your forum config was correct), I don't see how this would be happening.
It's also worth asking, if all the above is fine,
what version of MyBB are you trying to integrate? It's best with v1.6, which is slightly older. (Other types of forum software will not work, period, unless you write your own integration technique and cookie forwarder.)