I had this exact problem, and thought I'd post how it worked for me in the hopes it might save some TCHers a bit of messing around. It's actually really simple once you know the steps.
First of all, ftp down all your phpBB2 directories from the old host, and then upload them to TCH.
Then back up the MySQL database on the old host (mine used webmin, not cpanel/phpMyAdmin, but it still worked fine), then import the database using phpMyAdmin on the new host. Create your database first on the new host, then add a user and connect it to the databse. Make a note of the names because they get converted, i.e. if your username is foo, and you make a database or user called foobar, you'll get foo_foobar.
BIG IMPORTANT PART - you MUST open the database backup and look for any line with a bunch of dashes and either remove it, or put a space after the first two dashes, i.e.
---------------------
becomes
-- -------------------
Not sure why this happens, but you need to add the space to turn that line into a commment, which will be ignored, rather than a line of data, which will choke the import.
So, now import your cleaned up data into your new database using phpMyAdmin (open the query window, then hit 'import files').
Now the only thing you should need to do is open your local copy of config.php, open in a text editor, ignore the lines that say "Do not change anything in this file' and edit $dbname, $dbuser (and $dbpassword, if you changed it), to match the new names, and upload to TCH, overwriting the one on the server.
You're done!
PS - I used a very similar procedure to move over my LARGE Movable Type blog with a minimum of fuss. Now just trying to figure out why my mtamazon plugin isn't cooperating.
PPS - this is what worked for me, I take no responsibility whatsoever if you destroy your database, accidentally email your entire pr0n collection to everyone in your address book, or blow yourself up.