nohup - permission denied

You may have been misunderstanding sudo a bit — sudo does not necessarily elevate a user's rights; in fact, it may reduce the rights they have. When you pass sudo the -u flag, it will change to that user. If that user has more rights — root, for example (the default if -u is not passed) — then you'll get more rights. If the user has less rights — nobody, for example — you'll have less rights. You said that the log directory has these permissions:

drwxrwxr-x 2 root root 4096 Dec 11 10:13 /usr/local/orientdb/log

Yet, you're changing to the www-user user. The www-user user, unless it is part of the root group (unlikely), will not be able to write to that directory: it is only writable by the owner and group, and www-user is clearly not the root user and www-user is probably not part of the root group.

In short, don't pass -u (and its associated argument) if you want to elevate to root.

Try rewriting the last line of your script to:

sudo -u $ORIENTDB_USER sh -c "/usr/bin/nohup \"$ORIENTDB_DIR\"/bin/server.sh 1> \"$ORIENTDB_DIR\"/log/orientdb.log 2> \"$ORIENTDB_DIR\"/log/orientdb.err &"

If that still doesn't work, then www-user probably doesn't have write permission to /usr/local/orientdb/log (Note that you said /usr/local/orientdb/logs exists but is an empty folder: one of them has a s at the end)