I use for web hosting, and havent had any problems with my sites, forums or any of the scripts. In fact, the service & support has been the best I've experienced in 8 years of trying various providers.
When you switch hosts you have to have a full backup of your forum, which you can do through the admin panel. Then you will install phpbb on your new host, and restore that backup.
The only problem I ran into when I moved was that my backup file was too large, and I couldnt get it to restore from my end. I tried breaking the file down into multiple files - unsuccessfully... and finally emailed support at ThirdSphere. They took the file and restored it from there end for me and I was up & running! I was impressed that they were willing to help with that, instead of saying "we dont support that".
ThirdSphere is a full-feature host, so it may seem a little pricey compared to others at 24.95/month. That said, you can host unlimited domains on the one account... so it's WELL worth it if you have several sites Not to mention they have a built in affiliate program manager and other features that cost you a pretty penny elsewhere.