I had the same problem but found a way around it.
MyBB does not allow PHP code in the template system.
Only HTML. You could try this plugin first
http://community.mybboard.net/thread-31860.html , but if that doesn't work then you could try this other way.
It's not easy but it works. In every file such as index.php, usercp.php, misc.php etc. (all the files that are visible to the user) look for
all the "output_page" functions. It will look something like this
eval("\$index = \"".$templates->get("index")."\";");
output_page($index);
and add your php includes as needed like this, surrounding the entire output_page and eval.
include 'header.php';
eval("\$index = \"".$templates->get("index")."\";");
output_page($index);
include 'footer.php';
Also note that there can be 10+ of these in one file so that means lots and lots of copying and pasting. You just have to come up with a systematic way of doing it and you can get it done in a day.
Then delete the unnecessary elements from your MyBB header template as you no longer need alot of that. Just keep the welcome block stuff and get rid of any header images and things.
Assuming your header.php ends with something like <div id="content"> and your footer.php begins with the corresponding end div </div>
then all the forum content will get sandwiched in there.
HEADER.PHP STUFF
<div id="content">
MYBB FORUM WILL BE HERE
</div>
FOOTER.PHP STUFF
I will warn you that this takes lots of time to do, and is not necessarily the best route. It can cause headaches later on when trying to update MyBB as you cannot just upload all the new files over top of your modified versions. You would lose all your work. Thus you have to manually make the changes for each new version.
It also can take quite a bit of playing around with this, as your result may vary depending on your existing site design.
I would recommend doing this ONLY if your header and footer have PHP code in them. If it is just straight HTML in your header and footer then do like TimB. suggested and just simply copy and paste the HTML into the header and footer templates in MyBB.