Shared Linux servers should handle all these requirements. They should all be at (at least) PHP 5.3.5, with a program underway to upgrade to 5.6. MySQL and MySQLi are supported. json is supported. If getimagesize() is a standard function, it should be there. zlib and GD are definitely there, and I know Imagemagik has been supported in the past and presumably still is. I don't know about remote FTP and XML, but if they're fairly standard, they should be there.
VPS/cloud and dedicated servers are more under your control, so you should be able to add anything missing. Windows servers are pretty up to date, but I don't use any so I can't tell you for sure.
People have hosted phpBB at Lunarpages for a long time, and Softaculous offers phpBB 3.1 for installation on LP shared servers, so I would assume it's still supported. Why don't you look at the hosting plans at lunarpages.com, and then email firstname.lastname@example.org to inquire about specific plans' technical details (ability to host phpBB). Get a commitment for a full refund if it turns out your server can't support phpBB, but I would be very surprised if that is the case.