You may see these "permission denied" errors even though named is running as root.
On Linux, BIND 9 drops most of its root privileges on startup. This includes the privilege to open files owned by other users. Therefore, if the server is running as root, the configuration files and zone files should also be owned by root. If the files must be owned by some other user, then the server should run as that user; this can be set by using the -u option on the command line when launching named.